The present invention relates to superconducting quantum circuits, and more specifically, to reducing spontaneous emission in circuit quantum electrodynamics by a combined readout and filter technique.
Superconducting quantum computing is a promising implementation of quantum information technology that involves nanofabricated superconducting circuits, using Josephson junctions as non-linear elements.
For an integrated circuit to behave quantum mechanically, the first requirement is the absence (or reduction) of dissipation. More specifically, all metallic parts need to be made out of a material that has minimal loss at the qubit operating temperature and at the qubit transition frequency. This is essential in order for electronic signals to be carried from one part of the chip to another without energy loss, which is a condition for the preservation of quantum coherence. Low temperature superconducting materials are utilized for this task, and accordingly quantum integrated circuit implementations are referred to as superconducting qubit devices.
According to one embodiment, a superconductor circuit is provided. The superconductor circuit includes a Δ (delta) circuit comprising a first node connecting a Purcell capacitor (CP) to a qubit coupling capacitor (Cq), a second node connecting the Purcell capacitor (CP) to a readout coupling capacitor (Cc), and a third node connecting the qubit coupling capacitor (Cq) to the readout coupling capacitor (Cc). A qubit is connected to the first node, and the qubit has a qubit frequency. A readout resonator is connected to the third node, and the readout resonator combines with the Purcell capacitor (CP) to form a filter. The second node is configured for connection to an external environment. Capacitance of the Purcell capacitor (CP) is determined as a factor of the qubit frequency of the qubit, such that the capacitance of the Purcell capacitor (CP) blocks emissions of the qubit at the qubit frequency to the external environment. The capacitance of the Purcell capacitor (CP) is set to cause destructive interference, between a first path containing the Purcell capacitor (CP) and a second path containing both the qubit coupling capacitor (Cq) and the readout coupling capacitor (Cc), at the qubit frequency in order to block the emissions of the qubit at the qubit frequency to the external environment.
According to one embodiment, a method of providing a superconductor circuit is provided. The method includes providing a Δ circuit comprising a first node connecting a Purcell capacitor (CP) to a qubit coupling capacitor (Cq), a second node connecting the Purcell capacitor (CP) to a readout coupling capacitor (Cc), and a third node connecting the qubit coupling capacitor (Cq) to the readout coupling capacitor (Cc). The method includes providing a qubit connected to the first node, wherein the qubit has a qubit frequency, and forming a readout resonator connected to the third node. The readout resonator combines with the Purcell capacitor (CP) to form a filter. The second node is configured for connection to an external environment. Capacitance of the Purcell capacitor (CP) is determined as a factor of the qubit frequency of the qubit, such that the capacitance of the Purcell capacitor (CP) blocks emissions of the qubit at the qubit frequency to the external environment. The capacitance of the Purcell capacitor (CP) is set to cause destructive interference, between a first path containing the Purcell capacitor (CP) and a second path containing both the qubit coupling capacitor (Cq) and the readout coupling capacitor (Cc), at the qubit frequency in order to block the emissions of the qubit at the qubit frequency to the external environment.
According to one embodiment, a superconductor circuit is provided. The superconductor circuit includes providing a Y circuit comprising a qubit connected to a qubit coupling capacitor (C′q), and the qubit has a qubit frequency. A combination readout resonator and filter includes a Purcell capacitor (C′P), and the Purcell capacitor (C′P) is connected to the qubit coupling capacitor (C′q). A readout coupling capacitor (C′c) is connected to the Purcell capacitor (C′P) and the qubit coupling capacitor (C′q), and the readout coupling capacitor (C′c) is configured for connection to an external environment. Capacitance of the Purcell capacitor (C′P) is determined as a factor of the qubit frequency of the qubit, such that the capacitance of the Purcell capacitor (C′P) blocks emissions of the qubit at the qubit frequency to the external environment. The capacitance of the Purcell capacitor (C′P) is set to short the combination readout resonator and filter including the Purcell capacitor (C′P) to ground at the qubit frequency in order to block the emissions of the qubit at the qubit frequency to the external environment.
According to one embodiment, a method for a superconductor circuit is provided. The method includes providing a Y circuit comprising a qubit connected to a qubit coupling capacitor (C′q), and the qubit has a qubit frequency. The method includes forming a combination readout resonator and filter including a Purcell capacitor (C′P), the Purcell capacitor (C′P) being connected to the qubit coupling capacitor (C′q), and providing a readout coupling capacitor (C′c) connected to the Purcell capacitor (C′P) and the qubit coupling capacitor (C′q). The readout coupling capacitor (C′c) is configured for connection to an external environment. Capacitance of the Purcell capacitor (C′P) is determined as a factor of the qubit frequency of the qubit, such that the capacitance of the Purcell capacitor (C′P) blocks emissions of the qubit at the qubit frequency to the external environment. The capacitance of the Purcell capacitor (C′P) is set to short the combination readout resonator and filter to ground at the qubit frequency in order to block the emissions of the qubit at the qubit frequency to the external environment.
According to one embodiment, a superconductor circuit is provided. The superconducting circuit includes a Δ circuit comprising a first node connecting a Purcell capacitor (CP) to a qubit coupling capacitor (Cq), a second node connecting the Purcell capacitor (CP) to a readout coupling capacitor (Cc), and a third node connecting the qubit coupling capacitor (Cq) to the readout coupling capacitor (Cc). A qubit is connected to the first node, and the qubit has a qubit frequency. A readout resonator is connected to the third node, and the readout resonator combines with the Purcell capacitor (CP) to form a filter. The second node is configured for connection to the external environment. An equivalent Y circuit of the Δ circuit includes a qubit coupling capacitor (C′q), a Purcell capacitor (C′P), and a readout coupling capacitor (C′c). Capacitances of the Purcell capacitor (CP), the qubit coupling capacitor (Cq), and the readout coupling capacitor (Cc) are determined by a Δ-Y transformation as follows:
Additional features and advantages are realized through the techniques of the present invention. Other embodiments and aspects of the invention are described in detail herein and are considered a part of the claimed invention. For a better understanding of the invention with the advantages and the features, refer to the description and to the drawings.
The subject matter which is regarded as the invention is particularly pointed out and distinctly claimed in the claims at the conclusion of the specification. The forgoing and other features, and advantages of the invention are apparent from the following detailed description taken in conjunction with the accompanying drawings in which:
The basic building block of quantum information is the quantum bit, or qubit. As unwanted interactions with the environment cause decoherence of the qubit, a delicate balance between isolating the qubit from its environment while still allowing for control and readout to be performed must be achieved. A common approach to mitigating between these competing desires is circuit quantum electrodynamics (cQED), in which a qubit is protected by, and readout is performed through, a resonator possessing a resonant frequency different from that of the qubit. By tuning the cross-talk between the qubit and the external environment in cQED, protection of the qubit can be achieved while still maintaining the ability to perform fast, high-fidelity readout.
Circuit quantum electrodynamics (cQED) is the solid-state analogue of cavity quantum electrodynamics, i.e., the study of the interaction between atoms and photons inside a high-Q cavity resonator. In cQED, atoms are substituted with superconducting qubits, which lie inside a superconducting transmission line resonator, in place of the 3D optical cavity. This architecture is promising to realize a quantum computer, with long coherence times and multi-qubit control. Here, embodiments consider modifications to the transmission line resonator used for readout that can provide qubit lifetime enhancements and increase measurement fidelity, while potentially occupying less space, essential for the scaling of this architecture.
Like atoms in excited states, superconducting qubits such as the transmon qubit spontaneously emit photons, thereby lowering the energy state of the qubit. Since quantum information is stored in the states of the qubits, this spontaneous emission of radiation represents a loss of information and is a major source of error in quantum information processing. The error rate for a quantum operation must be less than 10−4 for quantum computing to be feasible (10−2 for a surface code). However, in a resonant cavity only certain electromagnetic modes can propagate; therefore by placing a qubit in such a cavity, the decay rate can be modified, and this a phenomenon known as the Purcell effect. Dispersive coupling to a readout resonator, in which the state of the qubit can be read out through the cavity resonator's detuned frequency, also helps to suppress the qubit decay rate.
Note that for ease of understanding and not limitation, sub-headings are utilized below.
I. Dispersive Readout
Specifically, in implementing dispersive readout, a qubit with transition frequency co is coupled to a resonator with resonant frequency ωR such that the magnitude of the detuning ω−ωR is much greater than the coupling between the qubit and resonator. Additionally, it is desirable for the circuit to be in the strong coupling regime, where the coupling is greater than both the cavity decay rate κ and decoherence rate γ. (The total decoherence rate γ is related to the excited state relaxation rate γ1 and dephasing rate γφ by γ=γ1/2+γφ). The detuning is typically negative to avoid loss though higher-order harmonics of the resonator. The diagonalized effective Hamiltonian for the coupled system is
where σz is the Pauli matrix that measured the state of the qubit. Thus, the resonator frequency is ‘pulled’ by ±χ (known as the dispersive shift) depending on the state of the qubit. In this way, the state of the qubit can be read without destroying its state, known as a quantum non-demolition (QND) measurement. The state can be extracted by the transmitted photon number (if the measurement tone is at ωR±χ or phase (if the measurement tone is at ωR).
The qubit decay rate from the Purcell effect is given by
γη=[γ(ω)]/CΣ
where CΣ includes the Josephson capacitance of the qubit junction and all the shunting capacitances, and Y(ω) is the total admittance across the qubit junction, and co is the transition frequency of the qubit. In the limit where the qubit is capacitively coupled to a single mode resonator the Purcell effect is given by
where κ(ω)=ZenvZRCc2ω2 ωR is the loss rate of the resonator with impedance ZR and g=ωRCq√{square root over (ZRω/4CΣ)} is coupling between the resonator and qubit through capacitor Cq. Clearly, qubit lifetime can be increased by increasing the detuning. However, this comes at a loss of measurement fidelity, which scales inversely with detuning. The qubit-resonator coupling must be strong for dispersive readout, so lifetime cannot be increased by optimizing g. Additionally, for fast readout, K cannot be decreased. Given these tradeoffs, it is beneficial to explore other techniques of implementing dispersive readout while protecting the qubit.
II. Purcell Filter
A microwave element known as a Purcell filter can be introduced between the qubit and the outside world that increases qubit lifetime. Previous realizations of Purcell filters include quarter-wave open transmission line stubs and impedance transformations that mismatch qubit decay channels. By filtering at the qubit frequency co, but not at the cavity frequency ωR, the Purcell filter increases qubit lifetime without affecting the readout. Additionally, there is no need to sacrifice qubit-resonator coupling g or photon loss rate K, so fast high-fidelity measurements can be performed with enhanced qubit lifetimes. Embodiments present a scheme whereby the Purcell filter is combined with a readout resonator so that measurement and filtering are simultaneously performed.
Embodiments present a circuit that elegantly combines the functions of dispersive readout with that of a Purcell filter. This circuit may be implemented with any planar resonator structure, allowing flexibility in design. In particular, a quasi-lumped element resonator may take the place of the transmission-line resonators, offering considerable savings in substrate area.
III. Resonator Circuits
First, consider the effective impedance to ground, Zeff, of the element used for dispersive readout that lies between the qubit and the outside environment, as shown in
The aforementioned waveguide resonator used for dispersive readout can be modelled as the simple LC lumped-element oscillator of
(where the ∥ symbol indicates the circuit elements are combined in parallel) which has poles at ω=±ωR and a zero at ω=0. Similarly, the anti-resonator consisting of an inductor and capacitor in series, shown in
Here, the pole is at ω=0 and the zeroes are at ω=±ωR, and thus the filter has the exact opposite properties of the resonator, as expected. In particular, the anti-resonator rejects signals at the resonant frequency ωR, while the resonator passes at that frequency if connected to a matched load.
Embodiments are designed to combine both kinds of behavior in one circuit, where the readout is performed at the resonant frequency, and the filter is designed to reject the qubit frequency in order to prevent spontaneous emission. Consider the addition of a Purcell capacitor CP′ in series with the resonator, as in
which is seen to have poles at ω=0, ±ωR and zeroes when
so that the zero is necessarily less than ωR. A similar calculation for the circuit with the Purcell inductor LP′ in
which has poles at ±ωR and zeroes at
and the positive zero is necessarily greater than ωR. Note that since the impedance of the circuits in
The impedance to ground for the circuits discussed in
As noted above, the circuits (
IV. Interaction of Resonator with Qubit and External World
Putting the combined resonator and filter from the previous section into a superconducting circuit 300 containing a qubit and taking measurements from the outside world is diagrammed in
Performance parameters of the circuit 300 are calculated in order to model how the combined readout and filter 315 functions when implemented in an actual experiment. Qubit lifetime is calculated to be
and Zres=(jωLR)∥(1/jωCR). By analyzing the circuit 300 using network theory, the resonator-qubit coupling g is found to be
The readout rate κ is determined by the cavity Q by κ=ωR/Q and Q is estimated to be
where the effective resistance and capacitance are defined by
from the impedance seen by the resonator
After modeling the necessary capacitances to achieve the desired readout/filter characteristics, experimenters found that the Purcell capacitor C′P needs to be very large. Such a Purcell capacitor C′P may have very large stray effects or would need to be patterned using an additional dielectric step in fabrication, thereby risking other sensitive elements of the device. Instead, the Δ-Y transform allows embodiments to pattern the capacitors in the different configuration of
Once in this Δ-configuration circuit, maximum qubit lifetime is calculated and shown in
In
The measurement scheme can be used in conjunction with any planar readout resonator. For example, in use with a quarter-wavelength Purcell filter similar to that of, this scheme can broaden/enhance the protection at an eighth of the wavelength. Particularly, this scheme works with quasi-lumped resonators, which are much smaller than the transmission line resonators and Purcell filters. This is beneficial for scaling the cQED architecture to allow for multi-qubit coherent operations necessary in the realization of a quantum computer.
This filter 415 can be implemented easily with standard lithographic procedures for fabricating planar superconducting microwave devices. For example,
In
In this implementation, the qubit frequency is below the readout frequency when using C′P. However, when the qubit frequency is higher than the readout frequency, then the Purcell effect could be suppressed by coupling the readout resonator inductively via L′P as shown in
Now moving away from the sub-headings, as discussed herein, the value of the Purcell capacitance (C′P) is well approximated by just considering the zeroes of the impedance of the Purcell capacitor and resonator alone. At this point, the combined filter and resonator 315 is effectively a short to ground from a Y-configuration perspective. The zeroes of the impedance are given in the discussion herein as
where ωq is the frequency of the qubit 305,
is the frequency of the resonator 310, and
is the frequency of the anti-resonator. Solving this for C′P, the experimenters obtain the Purcell capacitance (in the Y-configuration) necessary to protect the qubit 305 as
This can then be converted to the CP used in design with the Y-Δ transformation as
and the other capacitors transform according to the equations discussed herein. Table 1 shows representative values of C′P and CP given qubit frequency f=ω/2π in
For the A-configuration,
For the Y-configuration,
At block 1010, a qubit 305 connected to the first node is provided, and the qubit 305 has a qubit frequency. At block 1015, the readout resonator 310 is connected to the third node, and the readout resonator 310 and the Purcell capacitor (CP) combine to form the filter 415 (i.e., the combination readout resonator and filter 415). By providing the qubit coupling capacitor (Cq) connecting the qubit 305 from the first node to the readout resonator 310 at the third node, the capacitance of the qubit coupling capacitor (Cq) determines the strength of the coupling between the qubit 305 and readout resonator 310.
At block 1020, the second node is to be connected to the external environment (Zenv). By providing the readout coupling capacitor (Cc) connecting the readout resonator 310 from the third node to the external environment at the second node, the capacitance of the readout coupling capacitor (Cc) determines the rate at which measurements may be made. As understood by one skilled in the art, the external environment (Zenv) includes measurement and control instruments.
At block 1025, the capacitance of the Purcell capacitor (CP) is determined (designed) as a factor of the qubit frequency of the qubit 305, such that the capacitance of the Purcell capacitor (CP) blocks emissions of the qubit 305 at the qubit frequency to the external environment (Zenv). At block 1030, the capacitance of the Purcell capacitor (CP) is set to cause destructive interference, between a first path containing the Purcell capacitor (CP) and a second path containing both the qubit coupling capacitor (Cq) and the readout coupling capacitor (Cc), at the qubit frequency in order to block the emissions of the qubit 305 at the qubit frequency to the external environment (Zenv). For example, in
The capacitance of the Purcell capacitor (CP) has a direct relationship to the qubit frequency of the qubit 305 in which the capacitance of the Purcell capacitor (CP) is to be increased when a value of the qubit frequency is increased in order to block the qubit frequency to the external environment.
The combination readout resonator 310 further possesses an inductance LR (which may represent a readout resonator inductor) and a capacitance CR (which may represent a readout resonator capacitor) connected in parallel together. The readout resonator 310 (connected to) has a relationship to the Purcell capacitor (CP) based on a Y-configuration (in
Upon a Δ-Y transformation to the Y-circuit 300 of
The readout resonator 310 provides dispersive readout of the qubit to the external environment. The combination readout resonator and filter 415 provides filtering at the qubit frequency of the qubit 305 to the external environment.
The capacitance of the Purcell capacitor (CP) is determined from factors that include the qubit frequency of the qubit 305, an inductance of the readout resonator LR, and a capacitance of the readout resonator CR such that the capacitance of the Purcell capacitor (CP) blocks emissions of the qubit at the qubit frequency to the external environment while allowing dispersive readout of a state of the qubit. Further, the Δ circuit Purcell capacitor (CP) then in turn also depends on the qubit coupling capacitor (Cq) and readout coupling capacitor (Cc) after the Y-circuit 300 in
The Δ circuit 400 has a relationship to a Y-configuration in circuit 300. The capacitance of the Purcell capacitors (CP and C′P) is determined based on the relationship between the Δ circuit and the Y circuit via the Δ-Y transformation.
The descriptions of the various embodiments of the present invention have been presented for purposes of illustration, but are not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, the practical application or technical improvement over technologies found in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein.
This application is a continuation of and claims priority from U.S. patent application Ser. No. 14/512,489, filed on Oct. 13, 2014, entitled “REDUCING SPONTANEOUS EMISSION IN CIRCUIT QUANTUM ELECTRODYNAMICS BY A COMBINED READOUT AND FILTER TECHNIQUE”, the entire contents of which are incorporated herein by reference.
This invention was made with Government support under W911NF-10-1-0324 awarded by Army Research Office (ARO). The Government has certain rights to this invention.
Number | Date | Country | |
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Parent | 14512489 | Oct 2014 | US |
Child | 15218548 | US |