Patent Grant
6919579
1. Field of the Invention
This invention relates to quantum computing and, more specifically, to solid state quantum computing qubits with superconducting materials.
2. Discussion of Related Art
Research on what is now called quantum computing traces back to Richard Feynman, See, e.g., R. Feynman, Int. J. Theor. Phys., 21, 467-488 (1982). Feynman noted that quantum systems are inherently difficult to simulate with conventional computers but that observing the evolution of a quantum system could provide a much faster way to solve some computational problems. In particular, solving a theory for the behavior of a quantum system commonly involves solving a differential equation related to the Hamiltonian of the quantum system. Observing the behavior of the quantum system provides information regarding the solutions to the equation.
Further efforts in quantum computing were initially concentrated on “software development” or building of the formal theory of quantum computing. Software development for quantum computing involves attempting to set the Hamiltonian of a quantum system to correspond to a problem requiring solution. Milestones in these efforts were the discoveries of the Shor and Grover algorithms. See, e.g., P. Shor, SIAM J. of Comput., 26:5, 1484-1509 (1997); L. Grover, Proc. 28th STOC, 212-219 (1996); and A. Kitaev, LANL preprint quant-ph/9511026 (1995). In particular, the Shor algorithm permits a quantum computer to factorize natural numbers. Showing that fault-tolerant quantum computation is theoretically possible opened the way for attempts at practical realizations of quantum computers. See, e.g., E. Knill, R. Laflamme, and W. Zurek, Science, 279, p. 342 (1998).
One proposed application of a quantum computer is the factoring of large numbers. In such an application, a quantum computer could render obsolete all existing encryption schemes that use the “public key” method. In another application, quantum computers (or even a smaller scale device such as a quantum repeater) could enable absolutely safe communication channels where a message, in principle, cannot be intercepted without being destroyed in the process. See, e.g., H. J. Briegel et al., LANL preprint quant-ph/9803056 (1998) and the references therein.
Quantum computing generally involves initializing the states of N qubits (quantum bits), creating controlled entanglements among the N qubits, allowing the quantum states of the qubit quantum system to evolve under the influence of the entanglements, and reading the qubits after they have evolved. A qubit quantum system is conventionally a system having two degenerate quantum states, where the state of the qubit quantum system can have non-zero probability of being found in either degenerate state. Thus, N qubit quantum systems can define an initial state that is a combination of 2N states. The entanglements between qubits and the interactions between the qubits and external influences control the evolution of the distinguishable quantum states and define calculations that the evolution of the quantum states perform. This evolution, in effect, can perform 2N simultaneous calculations. Reading the qubits after evolution is complete determines the states of the qubit quantum systems and the results of the calculations.
Several physical systems have been proposed for the qubits in a quantum computer. One system uses chemicals having degenerate nuclear spin states, see U.S. Pat. No. 5,917,322, “Method and Apparatus for Quantum Information Processing”, to N. Gershenfeld and I. Chuang. Nuclear magnetic resonance (NMR) techniques can read the spin states. These systems have successfully implemented a search algorithm, see, e.g., J. A. Jones, M. Mosca, and R. H. Hansen “Implementation of a Quantum Search Algorithm on a Quantum Computer,” Nature, 393, 344-346 (1998) and the references therein, and a number ordering algorithm, see, e.g., Lieven M. K. Vandersypen, Matthias Steffen, Gregory Breyta, Costantino S. Yannoni, Richard Cleve and Isaac L. Chuang, “Experimental Realization of Order-Finding with a Quantum Computer,” LANL preprint quant-ph/0007017 (2000), Phys. Rev. Lett., Vol. 85, No. 25, 5452-55 (2000) and the references therein. The number ordering algorithm is related to the quantum Fourier transform, an essential element of both Shor's algorithm for factoring of a natural number and Grover's Search Algorithm for searching unsorted databases, see T. F. Havel, S. S. Somaroo, C.-H. Tseng, and D. G. Cory, “Principles and Demonstrations of Quantum Information Processing by NMR Spectroscopy, 2000,” LANL preprint quant-ph/9812086 V2 (1999), and the references therein. However, efforts to expand such systems to a commercially useful number of qubits face difficult challenges.
Another physical system for implementing a qubit includes a superconducting reservoir, a superconducting island, and a dirty Josephson junction that can transmit a Cooper pair (of electrons) from the reservoir into the island. The island has two degenerate states. One state is electrically neutral, but the other state has an extra Cooper pair on the island. A problem with this system is that the charge of the island in the state having the extra Cooper pair causes long range electric interactions that interfere with the coherence of the state of the qubit. The electric interactions can force the island into a state that definitely has or lacks an extra Cooper pair. Accordingly, the electric interactions can end the evolution of the state before calculations are complete or qubits are read. This phenomenon is commonly referred to as collapsing the wavefunction, loss of coherence, or decoherence. See Y. Nakamura, Yu. A. Pashkin and J. S. Tsai “Coherent Control of Macroscopic Quantum States in a Single-Cooper-Pair Box,” Nature V. 398 No. 6730, P. 786-788 (1999), and the references therein.
Another physical system for implementing a qubit includes a radio frequency superconducting quantum interference device (RF-SQUID). See J. E. Mooij, T. P. Orlando, L. Levitov, Lin Tian, Caspar H. van der Wal, and Seth Lloyd, “Josephson Persistent-Current Qubit,” Science 285, 1036-39 (Aug. 13, 1999), and the references therein. The energy levels of the RF-SQUID correspond to differing amounts of magnetic flux threading the SQUID ring. Application of a static magnetic field normal to the SQUID ring may bring two of these energy levels, corresponding to different magnetic fluxes threading the ring, into resonance. Typically, external AC magnetic fields are also applied to pump the system into excited states so as to maximize the tunneling frequency between qubit basis states. A problem with this system is that the basis states used are not naturally degenerate and the required biasing field has to be extremely precise. This biasing is possible for one qubit, but with several qubits, this bias field fine-tuning becomes extremely difficult. Another problem is that the basis states used are typically not the ground states of the system, but higher energy states populated by external pumping. This requires the addition of an AC field generating device, whose frequency will differ for each qubit as the individual qubit parameters vary.
The race to create the first scalable, practical, and powerful solid state quantum computer has existed for over ten years. Ever since the notion of a quantum computer first became evident with Feynman in 1982, scientists have been creating qubits of various forms. There are currently a number of disclosed qubits, where the quantum states are realized in the doubly degenerate ground states of the flux in a superconducting loop. Inevitably, these qubits are only useful when controlled by magnetic fields, or by some other means which couple the qubit to the environment or provide other potential sources of decoherence. In order to overcome these sources of decoherence, a large amount of overhead is required to control and harvest the quantum power available from the qubit. However, the means by which this can be accomplished has as yet eluded scientists. Thus, there is a need for a qubit which does not require the coupling magnetic fields, but which can be controlled by applying and reading currents and voltages.
There therefore exists a need for integrated solid state structures that can form the basic building blocks out of which integrated circuits using quantum effects can be built. The desired structures are such that they can be read from, written to and operated on in an efficient and scalable manner.
In accordance with the present invention, a qubit is comprised of a multi-terminal junction, where two of the terminals of the junction are directly connected together, thus forming a superconducting loop. The superconducting loop introduces a phase shift so that the phase of the superconducting order parameter Ψ is shifted by απ in transition through the structure, where α ranges from −1, through zero (no phase shift), to 1. A phase shift can be produced, for example, by the inclusion of a phase shifter in the superconducting loop or by external application of a magnetic field through the superconducting loop.
A qubit according to the present invention can be constructed from a multi-terminal Josephson junction in which at least two terminals of the junction are coupled to a superconducting loop to form a superconducting loop and at least two further terminals are open and can be coupled to external current sources. The multi-terminal junction can be made of superconducting leads coupled, for example, by any combination of constriction junctions (also referred to as micro-bridges), tunnel junctions, or semiconducting two dimensional electron gas structures within the physical location. In some embodiments of the qubit, the terminals of the multi-terminal junction are coupled in a physical location whose size is less than the size of the qubit.
In some embodiments of the invention, properties of both a symmetric junction and an asymmetric junction can be utilized. In a symmetric junction, a change in the direction of the transport current in the junction equally affects current in the terminals of the superconducting loop, thus having no overall affect on the current in the loop. In an asymmetric junction, a change in the direction of the transport current differentially affects the terminals that form the superconducting loop, thus changing the overall current in the loop.
A symmetric junction allows for the reduction of the potential energy barrier between the two nearly degenerate ground states of the quantum system of the qubit, thus providing a means of applying a σx quantum gate operation. An asymmetric junction allows for biasing of one of the two ground states of the qubit, thus providing a means of applying a σz quantum gate operation.
A phase shifter is any structure that shifts the phase of the superconducting order parameter Ψ by απ in transition through the structure, where α is a constant such that −1≦α≦1. The phase shift in the superconducting loop causes time-reversal symmetry breakdown in the qubit quantum system and thus causes a double degeneracy of the ground state without requiring an external magnetic flux or other influence. In some embodiments, the terminals in a multi-terminal junction can be physically asymmetric. This asymmetry affects the properties of a qubit according to the present invention by controlling the phase shift of the order parameter Ψ in transition through a multi-terminal junction.
A qubit according to the present invention may be constructed out of any superconducting material. Embodiments of qubits having any desired number of terminals and a phase shifter can also be constructed in accordance with desired applications for the qubit. Embodiments of qubit structures include, for example, s-wave superconductor/normal metal/d-wave superconductor/normal metal/s-wave superconductor, referred to as S-N-D-N-S junctions, superconductor/ferromagnet/superconductor, referred to as S-F-S junctions, s-wave superconductor/two dimensional electron gas/s-wave superconductor, referred to as S-2DEG-S junctions, or multi-crystal d-wave superconductors patterned on an insulating substrate. The equilibrium ground state of the qubit quantum system is, in the absence of external magnetic fields, twice degenerate, with one of the energy levels corresponding to a magnetic flux threading the loop in one sense (corresponding to an equilibrium supercurrent flow, for example, in the clockwise direction around the superconducting loop), and the other energy level corresponding to a magnetic flux threading the loop in the opposite sense (corresponding to an equilibrium supercurrent flow, for example, in the counterclockwise direction around the superconducting loop).
Some embodiments of qubits according to the present invention include an s-wave (for example, niobium, aluminum, lead, mercury, or tin) superconducting structure that includes an asymmetric four-terminal junction with all terminals connected by constriction junctions. Two of the terminals can be joined to form a superconducting loop and the other two terminals can be coupled to a source of transport current. The superconducting loop includes a phase shifter, which may consist of a S-N-D-N-S (for example, niobium/gold/YBa2Cu3O7-x/gold/niobium) junction. If the incoming current is parallel to the a (or b) crystallographic direction of the d-wave material, and the outgoing current is parallel to the b (or a) crystallographic direction of the d-wave material, this S-N-D-N-S junction can give a phase shift of π. Choosing the incoming and outgoing currents to be at any arbitrary angle to each other in the a-b plane in this embodiment allows a more general phase shift.
A magnetic field may also be applied to the superconducting loop. Both the transport current and the external magnetic field may be controlled so as to initialize the state of the qubit quantum system, allow control of the evolution of the qubit quantum system state and read the final state of the qubit quantum system after the evolution (and therefore the desired calculation) is complete. Further, qubits can be selectively entangled by coupling superconducting loops from different qubit structures with a switchable junction, allowing for control of entanglements in a qubit array.
A qubit according to the present invention can include a junction with any number of terminals. Some embodiments of the invention include a five terminal junction. A superconducting loop is formed between two terminals of the five terminal junction. The remaining three terminals, two terminals adjacent to the looping terminals, and one terminal centrally opposite the looping terminals, form a means by which to implement all desired quantum operations, including reading, writing, σx, gate operation, and a σz gate operation.
Some embodiments, such as the five-terminal qubit, include both symmetric and asymmetric properties. Because the critical current in the junction depends on the state of the qubit, a read operation can be performed by applying a current asymmetrically across the junction, with a magnitude between the critical currents of the two states and determining if a resistance is created. Additionally, a σz gate operation can be performed by applying a pulse of current asymmetrically across the junction while a σx gate operation can be performed by applying a pulse of current symmetrically across the junction.
In accordance with some embodiments of the invention, a quantum computing method cools a structure containing at least one multi-terminal qubit to a temperature that makes the structure superconducting and suppresses decoherence processes in the system. The actual temperature will depend on the superconducting materials of the qubit. After the structure is at the appropriate temperature, a supercurrent can be established in each superconducting loop, the supercurrent being in a particular classical bit state corresponding to the information input for the quantum calculation. The quantum systems of each of the plurality of qubits is then allowed to evolve in the presence of externally applied magnetic fields and transport currents (whose details constitute the “software” or algorithm being followed by the structure). This allows each superconducting loop bit state to evolve into quantum states that are admixtures of a first state having a first magnetic moment and a second state having a second magnetic moment. These quantum states evolve under the action of the system's Hamiltonian in the manner prescribed by quantum mechanics. The evolution performs the quantum computation. Determining a measured magnetic moment or flux due to the supercurrent in each superconducting loop determines the result of the quantum computation.
In accordance with another aspect of the invention, determining the measured magnetic moments of the quantum state on the qubit can also include applying an alternating transport current and/or magnetic bias field to each qubit and then measuring the magnetic flux produced by the supercurrent flowing in each superconducting loop. In some embodiments, a static transport current and/or magnetic bias field can be applied to each qubit and the voltage across at least two of the terminals in the multi-terminal junction measured to determine the quantum state of the qubit. In some embodiments, the quantum states of the qubits can be read directly with, for example, a SQUID magnetometer.
In further aspects of the invention, quantum qubits can be selectively entangled with switchable junctions. In embodiments where qubits include a superconducting loop, an array of qubits can be entangled by switchably coupling the superconducting loops of the array. Additionally, a switchable junction can be included to decouple selected ones of the superconducting loops from other multi-terminal junctions.
These and other embodiments according to the present invention are further discussed below with respect to the following figures.
a and 10b illustrate a switchable entanglement method for coupling two five terminal qubits.
FIG. 15. Mesoscopic 4-terminal SQUID qubit. The π-phase shifter in the flux loop is added to attain bistability without external flux. The externally controlled transport current I affects the current J in the superconducting loop through the phase dragging effect and in turns in the loop.
FIG. 16. Two coupled five-terminal qubits. The left qubit has an asymmetric current configuration while the right one has a symmetric one.
FIG. 17. System of N coupled qubits. The distance between the flux regions is maximized to have less magnetic interaction between qubits.
Use of the same reference symbols in different figures indicates elements having similar or identical functions.
In accordance with embodiments of the invention, a quantum computing operation can be performed on an array of quantum qubits where at least one of the qubits includes a qubit according to the present invention. A qubit according to the present invention includes a multi-terminal junction where two terminals of the multi-terminal junction are joined to form a superconducting loop. The superconducting loop introduces a phase shift to the superconducting order parameter. In some embodiments, the superconducting loop includes an intrinsic phase shifter.
Intrinsic phase shifters in superconducting phase quantum bits (qubits) are disclosed in M. H. S. Amin, T. Duty, A. Omelyanchouk, G. Rose and A. Zagoskin, U.S. Provisional Application Ser. No. 60/257624, “Intrinsic Phase Shifter as an Element of a Superconducting Phase Quantum Bit,” filed Dec. 22, 2000, herein incorporated by reference in its entirety. A phase shifting structure with 0 and π-phase shifts in a two-terminal DC SQUID is described in R. R. Schulz, B. Chesca, B. Goetz, C. W. Schneider, A. Schmehl, H. Bielefeldt, H. Hilgenkamp, J. Mannhart and C. C. Tsuei, “Design and Realization of an all d-Wave dc π-Superconducting Quantum Interference Device,” Appl. Phys. Lett. 76, 7 p. 912-14 (2000), herein incorporated by reference in its entirety.
Embodiments of qubits 100 according to the present invention can include phase shifter 123, which introduces an arbitrary phase shift inclusively between −π and π. Phase shifters, such as phase shifter 123, that introduce arbitrary phase shifts can be more practical for the construction of qubits since the phase shifters with arbitrary phase shifts are more easily constructed. Further tuning of the phase shift accumulated through superconducting loop 122 in qubit 100 can be accomplished by the application of a magnetic field or adjustment of the transport current IT in superconducting loop 122.
Additionally, embodiments of the present invention include at least one terminal junction 120. Terminal junction 120 joins at least two terminals, terminals 110-1 through 110-N. In some embodiments, the physical size of junction 120 is much less than the size of superconduction loop 122.
Four-terminal SQUID devices are discussed in A. N. Omelyanchouk and Malek Zareyan, “Ballistic Four-Terminal Josephson Junction: Bistable States and Magnetic Flux Transfer”, Los Alamos preprint cond-mat/9905139, and B. J. Vleeming, “The Four-Terminal SQUID”, Ph.D. Dissertation, Leiden University, The Netherlands, 1998, both of which are herein incorporated by reference in their entirety. Four terminal SQUID devices are further discussed in R. de Bruyn Ouboter and A. N. Omelyanchouk, “Macroscopic Quantum Interference Effects in Superconducting Multiterminal Structures”, Superlattices and Microstructures, Vol. 25 No 5/6 (1999), herein incorporated by reference in its entirety.
A quantum computation relies on qubit 100 including a qubit quantum system formed by supercurrents on superconducting loop 122 having degenerate ground states, designated |0> and |1>, of the supercurrent. An array of multi-terminal superconducting loops 122 having phase shifters 123 can be fabricated in useful numbers in a solid state structure. The ground state of the qubit quantum system includes two states that correspond to supercurrent flows that circulate clockwise and counterclockwise, respectively, in the plane of superconducting loop 122. The qubit quantum system of qubit 100 can be initialized by the introduction of supercurrents from an external source some of terminals 110-1 through 110-N not coupled to superconducting loop 122. The ground-state of the qubit quantum system in each superconducting loop 122 containing phase shifter 123 is doubly degenerate in the absence of externally applied magnetic fields and/or transport currents IT (each circulation direction has the same energy) and provides the basis for a qubit 100 for quantum computing in accordance with embodiments of the present invention.
The two degenerate states, corresponding to classical bit states, represented symbolically as |0> and |1>, are then the two basis states of the qubit quantum system of qubit 100. The magnitude of the flux threading superconducting loop 122 can be much less than half a flux quantum Φ0, both because of intrinsic phase shifter 123 and the presence of the terminals 110-1 through 110-N, which also introduces a phase shift. At least two external terminals, terminals 110-2 and 110-j in
The choice of the physical sizes of the constriction junctions of junction 120, tunnel junctions and/or semiconducting two-dimensional electron gas structures that couple the terminals, also affects the functioning of qubit 100. To achieve a small total flux in superconducting loop 122 (which is desirable for decreasing the decoherence rate) and maximum influence of the transport current IT on the properties of superconducting loop 122, in some embodiments the links in the transport loop (e.g., the current loop providing current to junction 120) are much wider than the ones in superconducting loop 122. A small residual flux exists because of spontaneous supercurrents, even in the absence of external fields. In those embodiments, the height of the potential energy barrier between the two degenerate quantum states of the qubit quantum system of qubit 100 will be affected most pronouncedly by the transport currents IT.
Multi-terminal junction 120 includes two important regimes: symmetric and asymmetric. Further, multi-terminal junction 120 can display symmetric, asymmetric, or a combination of symmetric and asymmetric properties. In a symmetric junction, a change in the direction of the transport current in the junction equally affects current in the terminals of the superconducting loop, thus having no overall affect on the current in the loop. Whereas, in an asymmetric junction, a change in the direction of the transport current differentially affects the terminals that form the superconducting loop, thus changing the overall current in the loop.
A symmetric junction can be used to reduce the potential energy barrier between the two nearly degenerate ground states of qubit 100, thus providing a means of applying a σx quantum gate operation. If a change in the direction of the transport current in terminals 110-2 through 110-j has an equal effect on the current in terminals 110-1 and 110-N, then junction 120 is symmetric with respect to terminals 110-2 and 110-j. Qubit 100, under these circumstances, can then be referred to as a “symmetric qubit”.
An asymmetric junction can be used to bias one of the two ground states of the qubit, thus providing a means of applying σz quantum gate operation. If a change in the direction of the transport current in terminals 110-2 through 110-j causes a differential change in the current in the loop terminals 110-1 and 110-N, then the junction is said to be asymmetric. Qubit 100, then, can be referred to as an “asymmetric qubit”.
Therefore, a four-terminal junction 120 can be either a symmetric or asymmetric junction, and four-terminal qubit 100 can then be either a symmetric or asymmetric qubit. If qubit 100 includes a junction 120 with more than four terminals, for example a five-terminal qubit, then both symmetric and asymmetric properties can be realized.
An asymmetric qubit can be written to (i.e., the quantum states initialized) by applying a transport current IT when the magnitude of the transport current IT is larger than a threshold value which is determined by the specific implementation of qubit 100. The direction of transport current IT is chosen depending on which basis state (i.e., |0> or |1> ) is being written into the qubit quantum system. The application of transport current IT, then, has the effect of biasing the qubit quantum system states into one of the degenerate basis states. In the biased state the qubit quantum system will decay to the most energetically favorable state (either |0> or |1> as required). In such systems, the time to decay typically is shorter than about 1 millisecond, depending on the particular embodiment of qubit 100. Depending on the particular embodiment of qubit 100, a magnetic field {right arrow over (B)} can also be applied in one of two directions, which can alter the time to decay. The magnetic field can be applied opposing the transport current induced bias, thereby decreasing the time for decay, or supporting the transport current induced bias, thereby increasing the time to decay.
A symmetric qubit can be written to (i.e., biased) by applying a static magnetic field {right arrow over (B)}. As described above, this will cause the qubit quantum system of qubit 100 to decay into the energetically favorable state on a time-scale dependent upon the embodiment of qubit 100 and the magnitude of an externally applied magnetic field {right arrow over (B)}.
Single qubit operations on asymmetric qubits can be performed by modulating the transport current and/or the external magnetic field strength. Setting the transport current IT to zero sets the effective Hamiltonian describing the quantum system of qubit 100 proportional to {circumflex over (σ)}x, which is referred to as a Pauli matrix. In the basis where the qubit basis states |0> and |1> are chosen so that the state |0> corresponds to the vector (1,0) and the state |1> corresponds to the vector (0,1),
This basis can be called the Z-diagonal basis. In this basis the Pauli matrix {circumflex over (σ)}x rotates one of the basis states into the other basis state (i.e., {circumflex over (σ)}x|0>=|1> and {circumflex over (σ)}x|1>=|0>).
Increasing the transport current IT past a threshold current, which can be an implementation dependent critical value, sets the Hamiltonian proportional to {circumflex over (σ)}z, which is another Pauli matrix. The matrix {circumflex over (σ)}z is defined in the Z-diagonal basis to be
The Pauli matrix {circumflex over (σ)}z biases the states, or in an alternative interpretation adds a phase to the second state (i.e. {circumflex over (σ)}z|0>=|0> and {circumflex over (σ)}z|1>=−|1>). An arbitrary single qubit operation can be performed by performing combinations of the functions described by {circumflex over (σ)}x and {circumflex over (σ)}z.
To keep the quantum system of qubit 100 in some specific state an alternating transport current IT(t) can be applied. The Hamiltonian representing the quantum system of qubit 100, then, is proportional to IT(t) {circumflex over (σ)}z. In some embodiments, for example, IT(t) can be a square wave. This method can be used in conjunction with a clock whose frequency is an integer multiple of the frequency of IT(t) so that, at every clock pulse, the quantum system of qubit 100 is in the same state in which it began. In addition, during the evolution of the qubit states external magnetic fields {right arrow over (B)} may be applied in accordance with a specific usage of the qubit.
Single qubit operations on symmetric embodiments of qubit 100 can be performed by modulating the transport current IT(t) and/or the external magnetic field {right arrow over (B)}. In the symmetric qubit embodiments of qubit 100, changing the transport current IT(t) does not affect the bias (i.e., the initial quantum states of the qubit quantum system) but does affect the energy barrier between the qubit quantum system basis states. The effective Hamiltonian describing the quantum system of qubit 100, then, includes a term proportional to ΔT(I){circumflex over (σ)}x where the tunneling matrix element ΔT(I) can be varied over a large range dependent on the transport current IT(t). Applying a magnetic field {right arrow over (B)} normal to the plane of the superconducting loop 122 provides another term of the Hamiltonian that is proportional to {circumflex over (σ)}z. To keep the quantum system in some specific state, an alternating magnetic field normal to the superconducting loop 122, {right arrow over (B)}(t), can be applied, adding a term proportional to {right arrow over (B)}(t){circumflex over (σ)}z to the Hamiltonian. In some embodiments, for example, {right arrow over (B)}(t) can be modulated in a square wave. This method can also be used in conjunction with a clock whose frequency is an integer multiple of the frequency of {right arrow over (B)}(t) so that at every clock pulse the qubit quantum system of qubit 100 is in the same state in which it began.
In some embodiments, the state of the qubit quantum system of an asymmetric embodiment of qubit 100 begins with the application of an alternating transport current IT(t) as described above. The magnetic flux threading the superconducting loop 122 can then be measured using a suitable magnetic field measuring device, for example a SQUID magnetometer or a magnetic force microscope. The alternating transport current IT(t) causes the decay rate out of the desired final state to be minimized. Once the state of the qubit quantum system of qubit 100 is measured, the transport current IT(t) can be set to bias the states such that the measured state has a lower energy level, making the measured state's information stable so that it may be accessed at any later time.
In some embodiments, the state of the qubit quantum system of the asymmetric qubit can be measured by applying a transport current IT(t) in a fixed direction and then monitoring the voltage drop across pairs of external leads, for example 110-2, 110-i and 110-j of FIG. 13. If the measured voltage remains constant then the state of the qubit quantum system corresponds to the favored energy state indicated by the fixed direction of the transport current IT(t). If the measured voltage changes, then the state of the qubit quantum system corresponds to an excited state indicated by the biasing of the fixed direction of the transport current IT(t). The changing voltage indicates a decay of the qubit quantum system state from the excited state. This indicates that the qubit quantum system was in the excited state relative to the bias indicated by the fixed direction of the transport current at the end of the calculation.
In some embodiments, the state of the quantum system of a symmetric qubit can be read similarly to the state of the quantum system of an asymmetric qubit, except that biasing is performed by application of magnetic fields normal to superconducting loop 122 and not via application of transport currents.
In an exemplary embodiment, four-terminal constriction junction 120 can be fabricated of aluminum. Widths W1 and W2 can each be approximately 0.5 microns; widths W3 and W4 can each be approximately 0.05 microns; lengths L1, L4, L5, L6, L7, and L8 can each be approximately 1 micron; and lengths L2 and L3 can each be approximately 0.55 microns.
In another exemplary embodiment, four-terminal constriction junction 120 can be fabricated of aluminum, with width W1 approximately 0.5 microns, W2 approximately 0.3 microns, W3 approximately 0.08 microns, and W4 approximately 0.05 microns, lengths L1, L4, L5, L6, and L7 each approximately 1 micron, L2 approximately 0.75 microns, L3 approximately 0.68 microns, and L8 approximately 0.9 microns.
In an exemplary embodiment, four-terminal junction 120 as shown in
Tunnel junction 140 can be fabricated where insulation layer 150 (
In another exemplary embodiment, four-terminal junction 120 as shown in
In an exemplary embodiment, four-terminal junction 120 as shown in
In another exemplary embodiment of junction 120 as shown in
In an exemplary embodiment, four-terminal junction 120 as shown in
In another exemplary embodiment of junction 120 as shown in
In an exemplary embodiment of junction 120 as shown in
In another exemplary embodiment of junction 120 as shown in
An exemplary embodiment of four-terminal junction 120 as shown in
Another exemplary embodiment of four-terminal junction 120 as shown in
In an exemplary embodiment, four-terminal junction 120 as shown in
In another exemplary embodiment, four-terminal junction 120 can be fabricated of niobium with the following dimensions: width W1 being approximately 0.05 microns; width W4 being approximately 0.08 microns; width W2 being approximately 0.1 microns; width W3 being approximately 0.15 microns; lengths L1. L2. L4. L5. L6. L7. and L9 being approximately 0.5 microns; and length Lbeing approximately 0.1 microns. The two dimensional electron gas structure of junction 140 can be formed of InAs.
An exemplary embodiment of four-terminal junction 120 can be fabricated of niobium with the following dimensions: widths W1 and W2 being approximately 0.5 microns; widths W3 and W4 being approximately 0.05 microns; lengths L1. L2. L4. L5. L6. L7. and L8 each being approximately 0.5 microns; and length D1 being approximately 0.1 microns. As previously discussed, the two dimensional electron gas structure of junction 140 can be formed of InAs. Additionally, tunnel junction 141 can be fabricated using a niobium oxide layer of thickness approximately 0.05 microns which is deposited onto terminal 110-2, upon which is deposited terminal 110-1.
Another exemplary embodiment of four-terminal junction 120 can be fabricated of niobium with the following dimensions: width W1 being approximately 0.05 microns; width W4 being approximately 0.08 microns; width W2 being approximately 0.05 microns; width W3 being approximately 0.15 microns; lengths L1. L2. L4. L5. L6. L7. and L8 each being approximately 0.5 microns; and length D1 being approximately 0.1 microns. The two dimensional electron gas structure of junction 140 can be formed of InAs and tunnel junction 141 can be formed with junctions 110-1 and 110-2 separated by a niobium oxide layer of approximately 0.05 microns thickness.
In general, multi-terminal junction 120 can include any number of terminals coupled by any types of junction. The embodiments of junction 120 shown in
Additionally, qubit 100 shown in
Modifying the angle of contact between the d-wave superconductor 240 and each of the two external terminals 210 and 211 modifies the phase shift acquired in transit through phase shifter 123 in a known way. For example, if the normal metal connectors, 250 and 251, are at a right angle to each other, the total phase is shifted by π through the phase shifter 123. Furthermore, if the normal metal connectors were directly opposite (0° apart), then there would be no accumulated phase shift. Following from this, any angle between 0° and 90°, where the angle is represented by θ, would lead to a phase shift of 2θ.
The physical characteristics of the normal metal connectors 250 and 251 can be chosen so as to provide standard Josephson junction connections between terminal 210 and d-wave superconductor 240 and terminal 211 and d-wave superconductor 240, respectively. Currents flowing in terminals 210 and 211 are labeled IS0 and IS1. respectively. The dimensions of d-wave superconductor 240 and connectors 250 and 251 are not critical.
In some embodiments terminals 210 and 211 can be niobium (Nb), aluminum (Al), lead (Pb) or tin (Sn). In some embodiments d-wave superconductor 240 can be YBa2Cu3O7-d. where 0.2<d<0.8. In accordance with an exemplary embodiment, terminals 210 and 211 can be made of niobium, connectors 250 and 251 of gold, and d-wave superconductor 240 of YBa2Cu3O6.68. Lengths LS0. LS1. LS2. and LS3 can be approximately 0.5 microns, widths WS0 and WS1 can be approximately 0.5 microns, and connectors 250 and 251 can be approximately 0.05 microns thick. The embodiment of phase shifter 123 shown in
In some embodiments, grain boundary 260 can be created by using a bi-epitaxial method where a d-wave superconductor is deposited onto a substrate containing seed layers upon which the d-wave superconductor grows in a different crystallographic direction than on the substrate itself. In some embodiments the substrate can be an insulator such as strontium titanate and the seed layers can be CeO (cerium oxide) or MgO (magnesium oxide). See F. Tafuri, F. Carillo, F. Lombardi, F. Miletto Granoziou, F. Ricci, U. Scotti di Uccio, A. Barone, G. Testa, E. Sarnelli, J. R. Kirtley, “Feasibility of Biepitaxial YBa2Cu3O7-x Josephson Junctions for Fundamental Studies and Potential Circuit Implementation, Los Alamos preprint cond-mat/0010128. published Phys. Rev. B 62, 14431-38 (2000), which is herein incorporated by reference in its entirety.
In some embodiments normal metal connector 250 couples d-wave superconductor 241 to s-wave superconducting terminal 211. In some embodiments normal metal connector 251 couples d-wave superconductor 242 to s-wave superconducting terminal 210. In some embodiments normal metal connectors 250 and 251 can be gold (Au), silver (Ag), platinum (Pt), or any other normal metal substance; and s-wave superconducting terminals 210 and 211 can be aluminum (Al), niobium (Nb), or any other superconductor with s-wave pairing symmetry.
In some embodiments all lengths and widths LS0. LS1. LS2. LS3. WS0. and WS1 can all be different, in some embodiments each of the lengths can be less than about one micron. The physical characteristics of normal metal connectors 250 and 251 can be chosen so as to provide standard Josephson junction connections between terminals 210 and d-wave superconductor 241 and terminals 211 and d-wave superconductor 240, respectively. Currents flowing in terminals 210 and 211 are labeled IS0 and IS1. respectively. The dimensions of d-wave superconductor 240 and connectors 250 and 251 are not critical.
In accordance with an exemplary embodiment of phase shifter 123 as shown in
Modifying the geometry of the ferromagnetic region 275 can change the angle of the phase shift in a known way. In
In some embodiments terminals 210 and 211 can be niobium (Nb), aluminum (Al), lead (Pb), tin (Sn), or any other superconductor with s-wave pairing symmetry. In some embodiments insulating region 275 can be aluminum oxide (AlO2) or any other insulating material. In some embodiments ferromagnetic region 276 can be an alloy of copper and nickel (Cu:Ni) or any other ferromagnetic material. One method for fabricating an example of phase shifter 123 as shown in
Terminals 110-1, 110-2, 110-3, 110-4, 210, and 211 along with portions 124 and 125 can be made of any superconducting material compatible with the particular embodiments of phase shifter 123 and four-terminal junction 120. Superconducting currents I1. I2. I3 and I4 can exist in terminals 110-1, 110-2, 110-3 and 110-4, respectively. Widths WJ1 through WJ4 are constrained only by the requirement of compatibility with junction 120 and widths WS0 and WS1 are compatible with phase shifter 123. In some embodiments, widths W1 through W4. WJ1 through WJ4. and WS0 through WS6 can all be less than about 10 microns. Lengths L1. L2. DJ1. DJ2. DJ3. DJ4. DP0. and DP1 are all compatible with phase shifter 123 and four-terminal junction 120 and, in some embodiments, can be all less than about 10 microns. Superconducting loop 122 can be threaded by a magnetic flux Φ, which may contain contributions from a spontaneous supercurrent in the loop and externally applied magnetic fields.
Four-terminal junction 120 couples one end of each of the four terminals 110-1 through 110-4 by, for example, constriction junctions, tunnel junctions, two-dimensional electron gas structures, or combinations of these. The choice of the physical sizes of the elements in four-terminal junction 120 that couple the four terminals 110-1 through 110-4 also affects the function of qubit 100. To achieve a small magnetic flux Φ in superconducting loop 122 (which is desirable for coherence consideration) and maximum influence of the transport current IT=I1=I2 on the properties of superconducting loop 122, in some embodiments the links in the transport loop are much wider than the ones in superconducting loop 122, that is, WJ1 and WJ2 are much larger than WJ3 and WJ4. A small magnetic flux Φ can exist even in the absence of external magnetic fields because of spontaneous supercurrents. In those embodiments, the height of the potential energy barrier between the two degenerate quantum states of the qubit quantum system of qubit 100 will be affected most pronouncedly by the applied transport current IT=I1=I2.
SQUID loop 122 with intrinsic phase shifter 123 can provide a basic block for construction of qubit 100 but can also be utilized for demonstration of macroscopic quantum tunneling and incoherent quantum noise in a solid state system. As described further below, the macroscopic quantum tunneling in a set of independent four-terminal qubits 100 with intrinsic phase shifters 123 (i.e., with no entanglements between individual qubits) permits construction of a random number generator that generates random series with zero correlation between numbers in the random series.
Four-terminal qubit 100 with intrinsic phase shifter 123 includes a superconducting loop 122 linking two of terminals 110-1 through 110-4, terminals 110-4 and 110-3 in FIG. 3A. Four-terminal qubit 100 with intrinsic phase shifter 123 can be made superconducting by reducing the temperature of qubit 100 below the superconducting critical temperature Tc of all of the superconducting materials utilized in the formation of qubit 100. Four-terminal junction 120 may be either symmetric or asymmetric, as was discussed with respect to
Four-terminal qubit 100 with intrinsic phase shifter 123 can be formed on an insulating substrate such as, for example, strontium titanate or sapphire. (See
Widths W1 through WN. widths WJ1 through WJN. and widths WS0 through WS6 are only constrained by the requirement of compatibility with phase shifter 123 and N-terminal junction 124 and, in some embodiments, typically all less than about 10 microns. Terminals 110-1 through 110-N of qubit 100 as shown in
N-terminal junction 120 couples one end of each of terminals 110-1 through 10-N by, for example, constriction junctions, tunnel junctions, two-dimensional electron gas structures, or combinations of these. The choice of the physical sizes of the elements in N-terminal junction 120 also affects the function of qubit 100. To achieve a small magnetic flux Φ in superconducting loop 122 (which is desirable for coherence consideration) and maximum influence of the transport currents I1 through IN on the properties of superconducting loop 122, in some embodiments the terminals in the transport terminals (i.e., terminals 110-1 through 110-N that are not terminals 110-I and 110-K) are much wider than terminals 110-I and 110-K at junction 120, that is, WJ1 through WJN. excluding WJI and WJK are much larger than WJI and WJK. A small magnetic flux Φ can exist, even in the absence of external magnetic fields, because of spontaneous supercurrents. In those embodiments, the height of the potential energy barrier between the two degenerate quantum states of the qubit quantum system will be affected most pronouncedly by the applied transport currents I1 through IN.
N-terminal qubit 100 with intrinsic phase shifter 123 can provide a basic block for construction of a qubit but can also be utilized for demonstration of macroscopic quantum tunneling and incoherent quantum noise in a solid state system. As described further below, the macroscopic quantum tunneling in a set of independent N-terminal qubits with intrinsic phase shifters (i.e., with no entanglements between individual qubits) permits construction of a random number generator that generates random series with zero correlation between numbers in the random series.
N-terminal qubit 100 with intrinsic phase shifter 123 includes a superconducting loop 122 linking two of the N terminals, terminals 110-I and 110-K. N-terminal qubit 100 with intrinsic phase shifter 123 is made superconducting by reducing the temperature of qubit 100 below the superconducting critical temperature Tc of all of the superconducting materials in qubit 100. N-terminal junction 120 may be either symmetric or asymmetric. The superconducting materials from which qubit 100A is constructed are constrained only by the requirement of compatibility with phase shifter 123 and N-terminal junction 120 and otherwise may have any pairing symmetry. For example, materials used may be s-wave, for example, niobium or aluminum, or d-wave, such as a high-Tc cuprate, for example YBa2Cu3O7-x. or any superconducting material in which the Cooper pairs are in a state with non-zero orbital angular momentum.
N-terminal qubit 100 with intrinsic phase shifter 123 can be formed on an insulating substrate such as, for example, strontium titanate or sapphire. Phase shifter 123 may be any structure that shifts the phase of the superconducting order parameter in transition across the structure. Examples of embodiments of phase shifter 123 are shown in
Terminal 410 is coupled to one of the terminals of qubit 401-1. Another of the terminals of qubit 401-1 is coupled to a terminal of qubit 401-2. Each of qubits 401-2 through 401-N are coupled to a terminal of qubit 401-1 through 401-(N−1). Additionally, a second terminal of qubit 401-N is coupled to terminal 411. Connectors joining qubits 401-1 through 401-N can be made of any superconducting material compatible with qubits 401-1 through 401-N. Superconducting current IL can exist in the loop formed by joining terminals 410 and 411. Widths WL1 through WL(N+1) and WSL1 through WSL3 are constrained only by the requirement of compatibility with qubits 400-1 through 400-N and are typically all less than about 10 microns. Lengths LSL1 through LSL3 and LL1 through LL(N+1) are not critical but, in some embodiments, are typically all less than about 10 microns.
The magnetic flux Φ481. which threads the loop formed by joining terminals 410 and 411, may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields. The magnetic fluxes Φ480-1 through Φ480-N. which thread the loops in multi-terminal qubits 401-1 through 401-N, respectively, may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields.
In accordance with embodiments of aspects of the present invention, qubit array 400 as shown in
Therefore, measurement of the flux Φ481. for example by an external measuring instrument such as a magnetic force microscope, scanning SQUID microscope or scanning Hall probe, provides information on the quantum states of qubits 401-1 through 401-N. In addition, application of time-dependent external magnetic fields to the superconducting loops of SQUID qubits 401-1 through 401-N and to the loop formed by joining terminals 410 and 411 can function as an operating system to perform specific algorithms using the qubit register architecture of qubit array 400 as disclosed in FIG. 4A. The relation of the time-dependent magnetic field application and the algorithm performed is described in Appendix A.
Terminals 410 and 411 can be made of any superconducting material compatible with the choice of qubits 401-1 and 401-N. Connectors joining qubits 401-1 through 401-N can be made of any superconducting material compatible with the choice of qubits 401-1 through 401-N. Superconducting current IT. which can be present in terminals 410 and 411, typically arises from an external transport current source.
Widths WL1 through WL(N+1) and WSL1 through WSL3 are only constrained by the requirement of compatibility with qubits 401-1 through 401-N and, in some embodiments, are typically all less than about 10 microns. Lengths LL1 through LL(N+1). LSL1 and LSL3 are not critical but are typically all less than about 10 microns. The magnetic fluxes Φ480-1 through Φ480-N which thread the superconducting loops in qubits 401-1 through 401-N, respectively, may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields.
In accordance with an embodiment of an aspect of the invention, qubit array 400 of
Examples of embodiments of phase shifters 523-1 and 523-2 are shown as phase shifter 123 in
The superconducting loops of qubits 500 and 501 can be threaded by magnetic fluxes Φ580 and Φ581. respectively. Terminals 110-1 through 110-6 can be made of any superconducting material compatible with the choice of qubits 500 and 501. Superconducting current IT in terminals 110-1 and 110-4 typically arises from an external transport current source. The superconducting loops of qubits 500 and 501 carry superconducting currents IQ0 and IQ1. respectively. The physical dimensions of terminals 110-1 through 110-6 are constrained only by the requirement of compatibility with qubits 500 and 501 and, in some embodiments, are typically all less than about 10 microns. The magnetic fluxes Φ580 and Φ581 which thread the superconducting loops of qubits 500 and 501, respectively, may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields.
In accordance with an embodiment of an aspect of the invention, qubits 500 and 501 as shown in
The superconducting loop of qubits 600-1 through 600-(N+1) can be threaded by magnetic fluxes Φ680-1 through Φ680-(N+1). respectively. The superconducting loops of qubits 600-1 through 600-N can include any superconducting material compatible with junctions 601-1 through 601-N and phase shifters 602-1 through 602-(N+1). Terminals 1 and 4 of each of junctions 601-1 through 601-N can carry superconducting currents IT1 through ITN. which typically arise from external transport current sources. The superconducting loops of each of qubits 600-1 through 600-(N+1) can carry superconducting currents LSL-1 through LSL-(N+1). respectively. All length and width scales are constrained only by the requirement of compatibility with junctions 601-1 through 601-N and phase shifters 602-1 through 602-(N+1), as has been previously discussed. The magnetic fluxes Φ680-1 through Φ680-(N+1) which thread the loops in multi-terminal qubits 600 through 600-(N+1) respectively may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields.
In accordance with an embodiment of an aspect of the invention, qubit array 610 as shown in
Magnetic fluxes Φ680-1,1 through Φ680-(M),(N) can be embraced by superconducting loops contained in qubit registers 690-1 through 690-M. The magnetic fluxes Φ680-11 through Φ680-(M)(N)B which thread the superconducting loops in qubits 600-1 through 600-N of each of qubit registers 690-1 through 690-M, respectively, may contain contributions from a spontaneous supercurrent in the loop and contributions from externally applied magnetic fields.
In accordance with an embodiment of an aspect of the invention, the array of registers 690-1 through 690-M shown in
In accordance with another embodiment of an aspect of the present invention,
The single-electron transistor (SET) 709 can be made of any material that displays a Coulomb blockade effect, for example niobium, aluminum, lead, tin, and any high-temperature superconducting cuprate. P. Joyez, P. Lafarge, A. Filipe, D. Esteve, and M. H. Devoret, “Observation of Parity-Induced Suppression of Josephson Tunneling in the Superconducting Single Electron Transistor”, Phys. Rev. Letters. Vol. 72. No. 15, 2458-61 (Apr. 11, 1994), describes operation and manufacture of single electron transistors and is incorporated by reference herein in its entirety. SET 709 is placed in a high quality factor tank circuit 712 tuned to resonance. Tank circuit 712 includes inductor 707 and capacitor 708. Capacitor 708 is coupled across SET 709. A third terminal of SET 709 is coupled to electrode 710. A radio-frequency or microwave signal 704 is introduced into the circuit 712. The reflected signal 705 is a strong function of the voltage difference between electrode 710 and ground 711. Analysis of reflected signal 705 using established techniques allows measurement of the voltage difference between electrode 710 and ground 711.
Read-out of the state of the qubit quantum system may be done via the use of a single electron transistor (SET) 709 according to known procedures, for example, as described in R. J. Schoelkopf, P. Wahlgren, A. A. Kozhevnikov, P. Delsing, and D. E. Prober, “The Radio-Frequency Single-Electron Transistor (RF-SET): A Fast and Ultrasensitive Electrometer,” Science. Vol. 280, 1238-42 (May 22, 1998). SET 709 may be coupled to three devices (e.g., terminals 710, 711 and 712). An electron or Cooper pair can tunnel onto SET 709 when SET 709 is uncharged. However, SET 709 is small enough that once an electron or Cooper pair tunnels onto SET 709, the charging of SET 709 electrically repels and prevents further tunneling onto SET 709. A terminal 710 associated with SET 709 can change the voltage of SET 709 and de-tune tank circuit 712, changing the characteristics of the reflected wave 705.
In operation, in order to measure a current, for example one of currents IA1 through IA(M−1) or IB1 through IB(N−1) shown in
All structures in
Some embodiments of the invention allow all of the operations that are required for quantum computing to be done without the application of external magnetic fields. Operations such as read and initialization, as well as operating system gates such as application of Pauli operators σx and σz. and furthermore operations for maintaining coherence in the state of the qubit can be achieved, without the use of external magnetic fields.
In some embodiments, as illustrated in
Although qubit 100 in
The quantum state of the quantum system of qubit 100 of
A read operation on qubit 100 of
The quantum state of the quantum system of qubit 100 of
In one method of measuring the quantum state of the quantum system of qubit 100, controller 800 applies a transport current IT to junction 120 which is between the known upper and lower critical current values (i.e., between the values of the critical current IC for each of the quantum states). The upper and lower values of the critical current IC is dependent upon the particular embodiment of qubit 100. When the transport current IT is applied, if the system occupies the state associated with the lower critical current, then the transport current will have exceeded the critical current value of the junction, thus resulting in a junction resistance, and a corresponding voltage across the terminals (for example, between terminals 110-1 and 110-3 to which the transport current IT is applied). Alternatively, if the system occupies the high critical current state, no voltage across the terminals will result. Controller 800, then, can determine the quantum state of the quantum system of qubit 100 by monitoring the voltage across junction 120 while applying the transport current IT through junction 120. For example, by applying a transport current from terminal 110-1 through terminal 110-3 of qubit 100 of
A phase gate operation σz can be performed, for example, on qubit 100 of
A phase gate operation σx can be performed, for example, on qubit 100 of
Furthermore, it is possible to tune the tunneling frequency of the qubit by applying a steady state current in the same manner as that of the σx operation. This allows a tuning of the quantum overlap of the two degenerate ground states in the qubit to a desired range. This is useful in an array of qubits where the tunneling frequencies of some or all of the qubits vary. By tuning the tunneling frequency of each qubit, the array can be tuned into a uniform range of tunneling frequencies, thus allowing more predictable application of quantum algorithms in the array.
Tuning can be achieved for example on qubit 100 of
a and 10b show operation of plate 146 in opening and closing junction 145 and thereby entangling or isolating qubits 100-1 and 100-2 of array 900. Controller 800 is electrically coupled to plate 146 at a terminal so that controller 800 can apply a voltage to plate 146, which is capacitively coupled to junction 145. Junction 145 can be any superconducting junction, including a two-dimensional electron gas, tunneling junction, or constriction junctions.
As shown in
As shown in
Although array 900 of
In operation, qubits and qubit arrays according to the present invention (such as, for example, the embodiments discussed above) are cooled to a temperature well below the superconducting transition temperature T, of all superconducting materials utilized to fabricate the particular structures. In an exemplary embodiment, the structures described in
In accordance with current theoretical descriptions, for example, the Eliashberg theory of superconductivity (see, e.g., R. de Bruyn Ouboter and A. N. Omelyanchouk, “Macroscopic Quantum Interference Effects in Superconducting Multiterminal Structures,” Superlattices and Microstructures. Vol. 25. No. 5/6 (1999)) an order parameter T describes current flow in superconductors and phase differences in multi-terminal junctions. Multi-terminal qubit 100 with intrinsic phase shifter 123 as shown, for example, in
The role of phase shifter 123 in multi-terminal qubit 100 is to cause the two basis states of the qubit to be naturally degenerate. This is a major advantage over other qubit designs, for example that of Mooij et al., Science 285 1036. where it is necessary to apply a magnetic field in order to bring the basis states of the qubit quantum system into resonance (i.e., cause them to be degenerate). The magnetic field required by the system of Mooij et al., Science 285 1036. needs to be extremely finely tuned in order to maintain the resonance condition. This is because the precision to which the external field has to be tuned is approximately the tunneling amplitude ΔT between qubit basis states which is usually about 5 GHz and corresponds to a magnetic field precision of one part in about 106.
One application of embodiments of multi-terminal qubits according to the present invention is a quantum computational random number generator. As a random number generator, the quantum states of an array of qubit 610 as shown in
Qubits according to embodiments of aspects of the current invention may also alternatively be read by other readout devices such as a magnetic force microscope (MFM) tip, a superconducting quantum interferometer device (SQUID) loop, or a Hall probe device. The readout device measures the weak local magnetic fields that the spontaneous supercurrents (clockwise or counterclockwise) cause in the vicinity of the multi-terminal qubit. More particularly, the MFM scans a microscopic magnetized tip attached to a cantilever across the surface and measures deformation of the cantilever as the mechanical force that acts on the magnetized tip. Alternatively, a superconducting loop can detect the magnetic flux in the vicinity of the multi-terminal qubit. Alternatively, a Hall probe can detect the magnetic flux in the vicinity of the multi-terminal qubit. Another possible read out system may use a difference in the absorption of circularly polarized microwave radiation due to the clockwise or counterclockwise currents by a multi-terminal qubit.
The time required for a calculation and the interpretation of the read out results depends on the calculation performed. Such issues are the subject of many papers on quantum computing, for example P. Shor, “Introduction to Quantum Algorithms,” Los Alamos preprint server condmat/005003 (Apr. 29, 2000). The structures described herein can serve as components of quantum computing systems and also can implement any single qubit algorithm.
In general a controller can be electrically coupled to provide current to the terminals of each qubit 100 in an array of qubits (for example, controller 800 shown in
Although the invention has been described with reference to particular embodiments, the description is exemplary only and should not be considered limiting. One skilled in the art may recognize several obvious variations, which are intended to be within the scope and spirit of the present disclosure. One skilled in the art will recognize embodiments of other qubits according to the present invention which are within the scope of this disclosure. Various adaptations and combinations of features of the embodiments disclosed are within the scope of the invention. As such, the invention is limited only by the following claims.
M. H. S. Amin1. A. N. Omelyanchouk2. A. Blais3. A. M. Zagoskin1,4. G. Rose1 and T. Duty1
1 D-Wave Systems Inc., 320-1985 West Broadway, Vancouver, BC V6J 4 Y3. Canada
2 B. I. Verkin Institute for Low Temperature Physics and Engineering, Ukrainzan National Academy of Sciences, Lenin Ave. 47. Kharkov 310164. Ukraine
3 Centre de Recherche sur les propriété de matériau électronique and Départment de Physique, Université de Sherbrooke, Sherbrooke Québec, Canada, J1K 2R1
4 Physics and Astronomy Dept., The University of British Columbia, 6224 Agricultural Rd., Vancouver, B.C., V6T 1Z1. Canada
We propose a novel superconducting flux qubit design based on mesoscopic 4-terminal junctions. The logical states correspond to different orientations of persistent currents in superconducting loops. Magnitude of the corresponding fluxes is of the order of a fraction of the flux quantum minimizing considerably interactions with the environment. No external fluxes are necessary to attain bistability or to perform logical operations.
In the past twenty years, clear evidences have been gained that quantum information processing could offer significant advantages over classical information processing [1, 2]. In parallel, it was recognized that superconducting systems, and particularly SQUIDs, are good candidates for the observations of quantum phenomena at the macroscopic level [3]. A great body of experimental evidences has been accumulated to support this assertion.
In this paper, a novel qubit design using the phase degree of freedom is introduced. Similarly to the rf-SQUID [4] or to the design introduced in [5] the logical states are represented by persistent currents of different orientations flowing in a superconducting loop. In the present design logical operations are not performed by manipulations of external fluxes. This is, among others, an advantage of this design.
The paper is organized as follows. In Sec. II we present the layout of the multi-terminal phase qubit. In Sec. III the quantum dynamics of the flux degree of freedom in this device is studied and it is shown how it can be reduced to a two-level system (TLS). In the following sections, we go on showing how this TLS satisfies the requirements for quantum computation [6]: initialization and measurement in Sec. IV, a universal set of gates is given in Sec. V, VI and VII and finally estimates for decoherence are presented in Sec. VIII.
The basic constituent of the novel qubit design studied in this paper is the 4-terminal Josephson junction (
where Δ0 is the superconducting gap in the banks, the γij are geometry dependent coupling coefficients between the ith and jth terminals and γij=0. A pair of terminals are either current biased or closed by a superconducting loop (see below). As noted in
As discussed in Article I, coupling between the I and J circuits is characterized by the matrix
For multi-terminal junctions such that det(γcoup)=0. coupling between the currents I and J is local. Constriction junctions (
By connecting terminals 3 and 4 via a superconducting ring, a 4-terminal SQUID is realized.
In this configuration, the 4-terminal SQUID is reminiscent of the much studied rf-SQUID [9]. It is known that for rings of large enough inductance and under Φ0/2 externally applied flux (Φ0=h/2e, the flux quantum) the rf-SQ can be bistable [4]. The bistable states correspond to different orientations of persistent currents or equivalently to different magnetic flux enclosed by the ring. Quantum behavior of the flux degree of freedom in this system has been the focus of numerous experimental and theoretical investigations over the past twenty years, see [4] for reviews.
Bistability of the 4-terminal SQUID has also been the focus of theoretical investigations and has been confirmed experimentally [10]. For quantum computing purposes, a major advantage of the 4-terminal SQUID over the standard rf-SQUID is that bistability can be attained for arbitrarily small loop inductance. As a result, the enclosed flux is much smaller than for the rf-SQUID and weaker coupling to the environment is to be expected. Small inductances and thus small fluxes were the main advantage of the 3 junction design of reference [5].
As in the rf-SQUID or 3 junction case, an external flux of Φc=Φ0/2 is needed to get bistability (see appendix A). However, as will become clear in the next sections, this control parameter is not exploited to manipulate the flux (i.e. qubit) state. The external flux can thus be fixed to Φ0/2 for all qubits. This opens the possibility of replacing, as shown in
Let us now proceed with the description of the 4-terminal SQUID. For clarity of presentation, the bulk of the calculations is given in the appendix and only the main results are presented here. It is useful to introduce the new variables:
where φ is related to the flux Φ threading the ring through φ=2πΦ/Φ0 mod 2π. It is also us variable φ=φ+π which, physically, is the phase related to the self-generated flux on the ring in the presence of a π-phase shifter or, equivalently, a Φ0/2 external flux.
In mesoscopic four terminal junctions where the 2DEG layer has a square geometry, the coefficient γij are such that γ13=γ24 and γ14=γ23. As shown in the appendix, this constrains χ to take only one of two possible variables, 0 and π. We will thereon work under this condition but results can be generalized to the more general case. We define the new variables γ=(γ13+γ14)/γ12 and δ=(γ13−γ14)/γ12. The difference between local and nonlocal couplings lies in the value of δ, namely δ=0 for the local case but δ≠0 in the nonlocal case.
In appendix A, the free energy of the 4-terminal SQUID is derived for T=0 and near the superconducting critical temperature Tc. It turns out that the free energy has the same form in both cases with differences only in the values of their coefficients. We therefore expect the same behavior to hold at all intermediate temperatures as well. All discussions in the main body of this article are devoted to the T=0 limit keeping in mind that the arguments are more general and that the same expressions can be derived near Tc as well. Evidently, the limit T=0 is more pertinent for qubit operation where one should expect longer coherence times.
From (A18) (keeping only the φ dependent terms), the free energy for the flux loop can be written as
where =2πI0L/Φ0 is the dimensionless inductance with I0 the Josephson critical current between terminals 1 and 2 (see appendix A). Energy is expressed in units of the Josephson energy E0=I0/2e of the 1-2 loop and current in units of I0. The coefficient of the second term, ε(I), and φ0(I) are functions of the transport current I:
with φ0(0)=γ/√{square root over (2)} and ε0=2δ. The energy (4) has two local minima at φ=±φ0(I) with energy difference 2ε(I). These minima correspond to different orientations of the persistent current in the superconducting loop. It is clear from (4) that the system can be bistable even if the superconducting loop as a very small inductance L. As stressed earlier, small inductance results in a small separation between the minima and correspondingly a smaller coupling to the environment.
The main difference between locally and non-locally coupled 4-terminal junctions, for which δ=0 and ≠0 respective now becomes apparent. In the former case, one as ε0=0 and the minima are always degenerate. In this case (or when ε(I) is tuned to zero) the barrier height between the minima is given by
In the non-locally coupled case, the bias term can be used to break this degeneracy and make one of the minima more favorable than the other. Hence, bias energy and barrier height can be tuned by the externally applied transport current. In the following sections, we will show how these properties can be used for initialization, readout and manipulation of the qubit's state.
In the presence of a small capacitance C across the junction 1-3, the system's total energy is
where Q is the number of charges on the capacitor and C=E0C/(2e)2 the dimensionless capacitance. The system's Hamiltonian is obtained by replacing Q by the appropriate operator in φ-representation, Q→i∂/∂φ. In units of E0 one then obtains
This is equivalent to a particle of mass proportional to the capacitance C moving in the double well potential of equation (4). For small enough capacitance, this system can behave quantum mechanically.
Near the two minima of (4), the classical oscillation frequency, in units of E0/, is given by
If this frequency is much smaller than the barrier height, in other words if ω0<<Ub(I), then the low energy levels on both sides of the barrier can be approximated by
When the minima are degenerate, coherent tunneling between the right and left well is allowed. In this situation, degeneracy between the two lowest levels is lifted by the tunneling splitting.
For the potential (4), A is identically zero and
One then obtains the tunneling splitting
In the regime were the attempt frequency ω0 is much smaller than the barrier height, but much larger than tunneling splitting, energy bias ε(I) and temperature, truncation to the Hilbert space spanned by the system's two lowest energy levels is possible. In the basis of the ground states of the right and left well respectively (which will be denoted {|0>, |1>} for obvious reasons), this yields the usual two level system Hamiltonian
Heff=Δ(I)σz−ε(I)σz. (15)
The σ's are Pauli matrices and the coefficients ε(I) and Δ(I) are given by (5) and (14) respectively. We have thus mapped the 4-terminal SQUID flux state's to a pseudospin. We will now proceed to show how this pseudospin can be used as a qubit.
Initialization is the process of taking an arbitrary state and bringing it to a well-defined state (which is usually chosen to be |0>). This evidently is a non-unitary process and is therefore possible only through coupling to an environment. Obviously, measurement is also a non-unitary process and also requires coupling to an environment (the measuring apparatus).
In this design, initialization and readout are performed through the phase dragging effect by applying a transport current. More specifically, for initialization a transport current I is applied for sufficiently long time. Because of the resulting energy bias (5), the system will relax to the lower energy state after sufficiently long time. By choosing the direction of current one can choose to whichever state to initialize to.
To readout the state of a qubit, one can use the fact that the critical current in the transport current loop depends on the state of the flux loop. As derived in appendix A, for the system to be bistable, θ (defined in (3) must satisfy the constrain |θ|<θmax. where θmax is given by (All). When this is satisfied, the current-phase relation in the transport current loop is given by (A15) and is reproduced here for convenience
The critical current therefore occurs near θ=π with the value Ic=1∓δ/2. depending on the qubit's state. In order to measure this state, one applies a current I=1. If the qubit is in the state +φ0. the current exceeds the critical current and the system goes to the non-stationary state with a voltage drop across the transport terminals. On the other hand if the system is the state −φ0. the current will be below the critical current and no voltage will be detected.
An important condition here is that when θ≈π the system should still preserve its bistable behavior. Therefore we need to have θmax>π. This imposes a minimum value for the equilibrium flux φ0(0)>4√{square root over (2)}(δ/γ), or equivalency puts an upper limit on δ
δ<γ2/8. (17)
Minimization of the flux φ0 while respecting (17) requires δ<<γ or equivalently γ13≈γ14.
Note that initialization can alternatively be performed by measuring the qubit's state. If the measurement result is |0>, initialization is complete. Otherwise, one can apply a π pulse (see below) to obtain |0> (this can be followed by a second measurement to cope with any imperfections in the pulse application).
Single qubit operations are generated by the single qubit Hamiltonian (15). As stressed in sections II and III, the coefficients ε(I) and Δ(I) of this effective Hamiltonian can be controlled through the transport current I. This gives the possibility to coherently manipulate the qubit's state.
Let us first consider locally-coupled multi-terminal qubits for which δ=0. In this situation, ε(I) is fixed to zero and only Δ(I) is externally controllable. For coherent manipulations, the control current I should not exceed the critical value Ic≈1 where the system becomes non-stationary. In practice, the maximum allowed value for I is slightly less than this critical value in order to avoid tunneling and thermally activated processes in θ-direction. Hence, currents roughly in the range [0, 1[ are useful for coherent control of Δ(I). In this range, the barrier height Ub(I) changes by a factor of ½. Since tunneling amplitude Δ(I) is exponentially dependent on this parameter, reducing Ub(I) by a factor of ½ can have a significant effect on the value of Δ(I). Thus, one can practically switch tunneling on and off by applying transport currents I=1 and I=0 respectively. As a result, when I=0. the qubit does not evolve, it is ‘frozen’. With a transport current I=1 however, the effective Hamiltonian is Δ(1)σx. and the qubit's evolution operator is
X(α)≡e−iσ
with α=2E0 Δ(1) t/ and where we reintroduced proper units for the tunneling splitting. Choosing t and I such that α=π, one obtains the π pulse of the previous section. Therefore, for δ=0. a qubit is either frozen (I=0) or undergoes coherent oscillations (I≈1).
In the asymmetric current configuration of
Z(β)=e−iσ
with β=2Eoε(I)t/.
It is well known that to implement arbitrary single qubit logical operations, one needs rotations around at least two orthogonal axis [12]. Since the situations δ=0 and ≠0 cannot be realized in the same 4-terminal SQUID we clearly have not yet reached this goal. Different ways to remedy this are possible. First, for δ≠0 the 4-terminal SQUIDs can be engineered such that for I=0 tunneling is not suppressed (e.g. by working with smaller capacitances). In this case, both X(α) and Z(β) operations are possible. It was shown in [13] how, in that situation, to ‘freeze’ a qubit evolution using NMR-like refocusing sequences. Alternatively, for δ=0. it will become clear in section VII that by using two-qubit gates and the single-qubit gate X(α), one can implement arbitrary single-qubit gates. Finally, we will see in the next section how the 4-terminal SQUID can be modified to effectively combine both the δ=0 and δ≠0 cases.
Before moving on, some comments are in order. First, for δ=0. when setting the transport current to I=0. one would expect the qubit to be frozen. However, since the tunneling amplitude Δ(0) is not identically zero, care must be taken to prevent residual coherent tunneling from introducing bit flip errors. Accordingly, we require that the time for a bit flip when the qubit should be frozen π/2E0Δ(0) be much larger than the time π/2E0Δ(1) when coherent oscillation are desired (when I≈1). For the error probability to satisfy the threshold condition for fault-tolerant quantum computation [2], one then roughly requires that Δ(0)<10−4Δ(1) or equivalently that Ub(0)>4n10ω0. which in turns puts a limit on the size of the capacitor
Moreover, it is important to make sure that the bias energy is always less than the difference between the energy of the lowest energy states and the first exited state. Otherwise, there might be leakage out of the computational Hilbert space causing loss of information [14]. To avoid this undesirable tunneling we require δ<ω0/√{square root over (2)}.
Finally, the double well potential (4) is obtained from (A18) which is a function of φ and θ. It is through relation (16) that θ, and thus Uφ. is a function of the transport current I. To obtain (4) however, the small term proportional to δ in (16) was dropped. For very small δ, this is reasonable but care should be taken because, through this term, θ does not only depend on the transport current I but also on the qubit's state. Since θ is not an observable quantity, this will not lead to additional environmental coupling. However, this effect, although very small can cause computational errors after a large number of operations. To minimize this, the junction parameter δ should then be chosen as small as possible. Since we are interested in junctions such that δ<<γ<<1. this corresponds to γ13. γ14<<γ12 and is not hard to achieve in practice. Note however that while this later requirement is in agreement with (17), junctions with very small δ's will require a correspondingly high level of accuracy in the applied transport current for measurement.
As exposed in the previous section, junctions with δ=0 and δ≠0 offer complementary possibilities for single bit operations and it would be practical to combine those possibilities in a single qubit design (this would for example remove the need of refocusing to freeze qubits in the asymmetric current configuration). This is impossible for the single 4-terminal SQUID because, as seen in
We suggest here to use the 5-terminal junctions illustrated in FIG. 16. Coupling between the two flux rings is used to create entanglement and will be discussed in the next section. The extra lead for transport current in this design gives the opportunity to use either symmetric (δ=0) or asymmetric (δ≠0) current configurations. More specifically, the qubit on the left hand side of
As a result, without transport current, the 5-terminal qubit is frozen (with appropriately chosen junction parameters). Applying a transport current I≈1 in the symmetric configuration a X(α) logical operation is performed while the asymmetric configuration yields a Z(β) operation. Arbitrary single qubit gates are then readily implementable in this design. This is done without manipulation of external fluxes but by injecting currents in appropriate leads.
Note finally that measurement and initialization should be performed here using the asymmetric current configuration.
To obtain a universal set of gates for the 5-terminal qubits, a prescription for a non-trivial qubit-qubit interaction is needed [12]. Here again, this interaction is provided by the phase dragging effect. Two qubits axe coupled through an additional mesoscopic 4-terminal junctions as presented schematically in FIG. 16.
The interaction can be turned off by applying a gate voltage to the 2DEG region between the two qubits. This voltage depletes the middle region and, as a result, prevents coherent coupling between the two flux loops. This interaction is studied in more details in appendix B and the resulting effective two-qubit Hamiltonian is
H2qb=H1+H2+Ω1,z{circle around (x)}σ2,z. (21)
where i=1, 2 indexes the qubits, Hi is the one-qubit Hamiltonian (15) and
The primed coefficients are parameters of the connecting 4-terminal junction (see appendix B). As indicated in the previous sections, the single qubits Hamiltonians can be manipulated individually through their transport current leads. When no transport current is applied on either qubits and when the 2DEG connecting the two qubits is not depleted, the system's evolution operator is then given by
CP(ζ)≡e−Iζσ
where ζ=2E0Ωt/ and CP stands for ‘Conditional Phase shift’.
As schematically shown in
CN12=X2(3π/2)CP12(π/2)Z2(π/2)X2(π/2)CP12(π/2)Z2(π/2)Z1(π/2). (24)
Some comments are in order. First, magnetic interaction between qubits should be very small to eliminate decoherence due to unwanted qubit-qubit interactions. The qubits should therefore be disposed in such a way as to keep the regions containing the maximum flux farthest apart. Secondly, it is possible, in this design, to apply logical operations simultaneously on distinct qubits. Since it is known that only quantum computers having this kind of classical parallelism can hope to satisfy a threshold condition for fault-tolerant quantum computation [2,15], this is a major advantage of this design. Note that not all (solid-state) quantum computer architectures have this ability [4].
Moreover,
As for any other qubit design, the most important limitation for performance is decoherence due to interaction of the qubits with the environment. Many sources of decoherence that may be active for superconducting flux qubits were discussed in [17]. see also [4]. These are fluctuation of background charges, quasiparticule tunneling in the superconductors, coupling of fluxes to nuclear spins, electromagnetic radiation and finally unwanted magnetic qubit-qubit couplings. At low temperature and for small equilibrium fluxes, coherence time was shown to be limited by the later source [17]. In the present design, this can be minimized as was discussed in the previous section.
Let us now consider additional sources that may cause decoherence in this system. One potential source of decoherence is the fluctuations of the external transport currents. These cause fluctuations of the tunneling matrix element Δ(I) and of the energy bias ε(I) and in turn affects the relaxation and dephasing time of the qubit [4,20]. The effects of fluctuations of a general external control field on the relaxation and dephasing times of a qubit are described in [4]. We will consider here fluctuations of transport current in the asymmetric configuration where they should affect dephasing and relaxation more strongly. In this situation, fluctuations of currents δI will provide a term of the form σzδI in the system+environment Hamiltonian. The power spectrum of the current fluctuations is given by Johnson-Nyquist relation
where Z(ω) is the impedance of the transport current circuit. For a purely real impedance, RI. the strength of qubit-environment coupling is characterized by the dimensionless parameter [4,21]
where RK=h/e2 is the quantum resistance. Hence, to minimize coupling, RI should be as large as possible and δ small. This last constraint goes along the constraints already imposed on δ in sections IV and V. From (26), one can estimate the relaxation and dephasing times respectively given by
where ΔE=√{square root over (ε(I)2+Δ(I)2)}{square root over (ε(I)2+Δ(I)2)} and Tcs is the temperature at which the current source circuit operates, which is not necessarily equal to the working temperature of the qubits (T). Tcs is usually higher than T. In the large Tcs limit, the two terms in (29) have the same functional form and we find
The relevant parameter for quantum computation is less the dephasing time by itself than the quality factor Q: the ratio of the dephasing time over the characteristic time it takes to implement a single logical operation. For the asymmetric current configuration this characteristic time is /E0δ and we obtain for the quality factor
For typical junction parameters, the Josephson energy E0=IO/2e is of the order of XX. For the corresponding frequency of operation of the qubit, the lead impedance is of the order of the vacuum impedance and thus RI˜100Ω[4,22]. Since δ<<1. for working temperatures in the milliKelvin range, the quality factor should be large. The ratio RI/RK˜10−3 is the limiting factor here. However, as we shall shortly see, this ratio could potentially be modulated.
The finite conductance of the normal quasiparticles through the 2DEG can also cause decoherence. The coupling of the quasiparticle current to the system is through the term Iqpφ0(I)σz. Following exactly the same procedure as above we find
where Rqp is the quasiparticle resistance of the junction (or shunt resistance in the Resistively Shunted Junction (RSJ) model). At small temperatures (kBT<<ΔE) we find
Although (32) does not show explicit T dependence at low temperatures, Rqp depends on temperature. As shown in [18]. the quasiparticle resistance is Rqp˜T cosh2(Es/2kBT), where ±Es are the energies of the Andreev bound states inside the normal region. Rqp exponentially diverges as T→0. Thus by working at low temperatures, this source of decoherence is made negligible. This is along the same line of reasoning as the discussion in [17] on decoherence due to quasi-particules tunneling in tunnel junctions although Es can be much smaller than the gap Δ0. which is the energy scale that comes into the exponent in tunnel junctions.
Moreover, care must be taken that the ac Josephson voltage generated by the phase fluctuations does not generate inter-Andreev level transitions [19]. This imposes ω0<Es which therefore requires that C>1/LE82. This constraint on the capacitance does not contradict the condition (20) already imposed in section V to minimize bit flip errors.
Pairs of qubits are also coupled to the electromagnetic environment through the gate voltage used to tune qubit-qubit interactions. When this interaction is turned off, i.e. when the coupling 4-terminal junction of
An additional source of dephasing is the spread in qubit parameters and the errors in gate applications that follows. This is common to all solid state architectures and can be taken into account, for example, by calibration [24].
Decoherence in this design is then limited by fluctuations of transport currents. With the estimated working frequencies, about XX logical operations can be performed within the dephasing time due to this source.
An advantage of this qubit design is that logical operations are not performed by manipulations of externally applied magnetic fluxes. For rf-SQUIDs and for the qubits introduced in [5]. independently controlled fluxes of the order of Φ0/2 must be applied to two different regions of each qubit. Given the size of those structures this is a technically challenging issue and imperfect shielding will cause dephasing.
Moreover, in the design studied here, since the flux is not used to manipulate qubit's state it can be replaced by π-phase shifters as discussed in section II. Inclusion of these elements can complicate the fabrication, as this will, for example, necessitate the use of d-wave superconductors. Alternatively, static external magnetic fluxes can be used. In this case however, fluctuations of the fluxes provide yet another coupling to the environment. As long as the spread in loop sizes is not to large, a single magnetic field can be applied to all qubits. In this case, the qubits are coupled to the same electromagnetic environment and, encoding two physical qubits in one logical qubit, the techniques of decoherence-free subspaces can be used [25,26]. These techniques where implemented successfully to protect quantum information from decoherence in situations where the qubit-environment coupling symmetry was the same as the one here [27,28].
Finally, one of the great challenges in the implementation of an useful quantum computing device is that coupling to the environment should be as weak as possible during coherent manipulations (i.e. computations) but should be as strong as possible during initialization. This later requirement comes, in part, from the fact that many error correction techniques requires continuous supply of fresh ancilla qubits. Therefore if initialization is slow (i.e. coupling to the environment is weak), the logical qubits for which error correction is intended will have had the time to decohere before fresh ancilla are prepared [6]. A possible solution is to initialize all the necessary qubits beforehand and distribute them in the quantum register as needed during computation [6]. This will however require very large quantum registers and many manipulations of qubit's state. Another possibility, is to tune the individual qubit's environment as needed either for computation or initialization. A possible way to implement this tunable environment is to use as transport current leads one-dimensional arrays of dc-SQUIDS as was done recently in [23]. By applying an external flux, the array act as a tunable electromagnetic environment where RI can be changed by many orders of magnitudes. Hence, during coherent manipulations using the asymmetric current configuration, one chooses a high resistance environment while during initialization, a small RI. If this can be done on individual qubit, one can then tune the decoherence time of individual qubits by many orders of magnitude.
We described in this paper novel superconducting flux qubits. These qubits are built from mesoscopic 4-terminal junctions. The logical states are represented by different orientations of persistent current in a superconducting loop. The corresponding fluxes differ only by a fraction of the flux quantum and therefore only weak coupling to the environment is to be expected. Logical operations are performed, through the phase dragging effect, by applications of transport currents. This compares favorably to other superconducting flux qubits [4,5,24]. where two independent controllable external fluxes must be coupled to each qubits.
Measurement is performed by applying a precisely tuned transport current in the control leads. As opposed to the other flux qubits, there is then no need to couple an additional dc-SQUID to the qubits and consequently simplifies fabrication and manipulations.
Decoherence should be slow in this system allowing for a large number of logical operations to be applied. To help in further reducing the impact of qubit-environment couplings, encoding physical qubits into decoherence free subspaces [25,26] or the use of active error correction techniques [2,29] is possible.
We start from the current-phase relationship (1) to derive the free energy for the four terminal SQUID. The expression (1) is simplified considerably near T=0 and near Tc. Near T=0 one finds
The Josephson coupling energy of the junction, EJ. which is related to the supercurrents Ii in Eq. (A1) through the relation Ii=(2e/)∂EJ/∂φi is given by
Near Tc on the other hand, one can write
with the Josephson coupling energy
In the subsections A and B we derive the free energy of a four terminal SQUID in these two limits. By comparing the results of the two subsections, one can easily see that the main properties are the same in the two extreme cases, T≈0 and T≈Tc. Therefore one can conclude that similar behavior should be expected at all other temperatures. For simplicity of calculations, in this article more focus on temperatures close to zero, which is the appropriate limit for qubit operation.
Considerable simplification in the calculation will be achieved if
γ13=γ24. γ14=γ23 (A5)
In that case X, as defined in (3), can only take two values X=0 or π (see subsections A and B). This condition naturally happens in a mesoscopic 4-terminal junction with square geometry.
At T=0 the Josephson energy of the 4-terminal junction will have the form
The energy is in units of E0=I0/2e, the Josephson energy of between terminals 1-2, and I0=πΔ0/2e is the Josephson critical current between those terminals at T=0. If we use the definitions of Eq. (3), the coupling part of the Josephson energy becomes
We use the simplifying assumption (A5) to get
In the presence of a π-junction or an external flux φe=π, we substitute φ→φ=φ−π
For small φ and θ
In that case, the minimum of the energy with respect to χ happens when χ=0 or π. Otherwise χ has to be close to π/2 and adds an additional undesirable factor of 1/√{square root over (2)}. Therefore we can write
Substituting γ=γ13+γ14 and δ=γ13−γ14 after some manipulations
When θ=0. the first (second) argument of Max is the largest for negative (positive) φ. Changing the sign of φ, the maximum switches from one argument to another. This results in two minima for the energy at two φ's with opposite signs. At finite θ, the two minima still exist as long as |θ|<θmax. where θmax is the value of θ at which the two terms inside Max become equal. θmax can be obtained by solving
where φ(θmax) is the equilibrium value of φ at θ=θmax. Using the minimum value for equilibrium φ we find
By making δ<<γ we can make θmax large enough so that we have bistability for all −π<θ<π. We then can write
Assuming small φ1 the free energy of a 4-terminal SQUID, can therefore be written as
We are interested in the limit where δ, κ<<γ<<1. Minimizing with respect to θ, we will have
The second term adds a small correction to the critical current which can be neglected. The last term on the other hand, changes sign from one minimum to another. This is important in the readout of the final state. The current-phase relation therefore can be written approximately as
The last term in (A15) can cause computation error when large number of operations is involved. Thus we want it to be small. Neglecting the last term we find
θ=2 sin−1(I) (A16)
The φ-dependent part of the free energy will then be
We can write this energy as
But now φ0(0)=γ/√{square root over (2)}, ε0=2δ and Ub(0)=φ0(0)2/2.
Near Tc. we simply write the Josephson energy as
Here the energy is in units of E0=I0/2e), the Josephson energy of between terminals 1-2, with I0=πγ12Δ02/4eTc being the Josephson critical current between those terminals. We also normalize all other γij to γ12. Introducing the new variables as in Eq.(3), we can write the Josephson energy of the junction in units of E0=I0/2e as
Assuming the condition (A5), and defining new parameters γ=γ13+γ14 and δ=γ13−γ14 we find
Minimization with respect to χ leads to χ=0 or π depending on the sign of the expression inside the parenthesis. This is directly consequence of (A5). In cases more general than (A5), χ can take any value (other than 0 and π) and the expressions become more complicated. We therefore find for the total energy of the 4-terminal junction
with κ=γ34. The free energy of a SQUID made of this 4-terminal junction is given by
Here, =2πI0L/Φ0=2eI0L/ is the dimensionless inductance of the ring. Now let's assume that an external magnetic flux φe=π is applied to the SQUID loop. Changing the variable φ→φ=φ−π, will give
Exactly the same equation is obtained if one adds an ideal π-junction in the SQUID-loop instead of applying the external flux. In that case φ will correspond to the actual flux treading the loop (φ=2πΦ/Φ0), and φ (=φ+π) will be the phase across the junction.
The interesting regime for qubit considerations is κ<<δ, γ<<1 and γ<<1. In this limit, the effect of the transport current on the qubit is maximal and the effect of the qubit in the current is minimal. The last condition insures that the difference between the flux produced in the two states is small and the qubits are so called “quite”; they do not interact with each other. The φ-dependent part of the free energy can then be written as
When the first term in the argument of the absolute value is the largest, the free energy remains unchanged when φ→−φ. In that case, the free energy has two minima at two values of φ=±φ0. Thus in order to achieve bistability the first term inside the absolute value should be the dominant one. This is possible only if |θ|<θmax where
Notice the similarity between this equation and (A11). At small φ we can write
Assuming that the sign of the argument inside the absolute value is determined by the first term, Eq. (A28) can be approximated by
We find two minima at φ±=±γcos(θ/2). The last term breaks the degeneracy between the two minima. The energies of the states become
The energy difference between the levels will be 2ε(θ), where
This term can provide σz operation for quantum calculations.
The phase θ is determined by the transport current I. From Eq. (A25), the minimization ∂U/∂θ=0 leads to
The second term in (A14) is the same at both minima. The third term on the other hand, changes sign from one minimum to the other. This small correction is important in the readout of the qubit state but it causes error in quantum calculations. Since γ, δ<<1. we can approximately write
θ=sin−1(I) (A33)
In terms of the current I, we can write the energy as
φ0. ε and the barrier height Ub(I) depend on the transport current I through
where φ0(0)=γ, ε0=2δ and Ub(0)=φ0(0)2/2. These expressions are exactly the same as what we found near T=0.
Let's consider two qubits coupled via a mesoscopic 4-terminal junction. Each qubit has its own control multi-terminal junction for manipulation and readout. The free energy of the coupled qubits is
EJ1 and EJ2 are the Josephson energies of qubits 1 and 2 respectively. The qubits have the same parameters κ, γ and δ. E12 is the Josephson energy of the coupling 4-terminal junction with parameters κ′, γ′ and δ′, and phase differences φ1′ and φ2′ on both sides of the junction. Taking φ1′, φ2′<<1. after simple algebra we find
If κ′>>γ>>γ′, δ′, then the state of the qubits will be dominantly determined by φ1 and φ2. In that case, we can ignore φi′ in all the calculations of the qubit states and the results previously obtained will hold. The value of φi′ then can be obtained from the equilibrium value of φt(=φi−π) by
where Ji is the current in the SQUID loop of the i-th qubit. The last term in (B2) give a coupling potential
In quantum regime the total Hamiltonian for the two qubits will therefore be
Htot=H1+H2+Hcoup (B5)
where
H2=Δ(Ii){circumflex over (σ)}i,x−ε(Ii){circumflex over (σ)}2,z.
Hcoup=Ω{circumflex over (σ)}1,z{circumflex over (σ)}2,z.
are the uncoupled Hamiltonian for the i-th qubit and Hcoup is the coupling Hamiltonian respectively.
This application is related to concurrently filed U.S. application Ser. No. 09/839,991 entitled “Quantum Bit with a Multi-Terminal Junction and Loop with a Phase Shift” and U.S. application Ser. No. 09/839,636 entitled “Quantum Bit with a Multi-Terminal Junction and Loop with a Phase Shift”, both of which are herein incorporated by reference in their entireties.
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