Patent Application
20110156008
This application claims benefit under 35 U.S.C. §119(e) of provisional U.S. patent application no. 61/290,661, filed Dec. 29, 2009, and of provisional U.S. patent application no. 61/290,645, filed Dec. 29, 2009.
This application is related to U.S. patent application no. [attorney docket MSRD-0032 (328991.02), entitled Ising Systems: Helical Band Geometry For DTC And Integration Of DTC Into A Universal Quantum Computational Protocol], filed on even date herewith.
The entire disclosure of each of the foregoing applications is incorporated herein by reference.
The subject matter disclosed and claimed herein relates generally to the field of quantum computing. Specifically, the subject matter disclosed and claimed herein relates to methods for implementing the π/8 gate in a genus=1 Ising system.
The term “Ising system” may be used herein to denote any physical system, particularly a two-dimensional electron gas (2DEG) whose low energy effective theory is governed by the Ising topological quantum field theory (TQFT). Below, the defining characteristics of the Ising TQFT are set out in tabular form using the notation and formalism of modular tensor categories (MTC). Although the structure appears to require much data, actually this direct description of properties is quite redundant: according to Kitaev, there are precisely eight theories with the Ising fusion rules and these can be distinguished by a Chern class.
A microscopic construction of the Ising system was given in Kitaev [arXiv:cond-mat/050506438v3], incorporated herein by reference, and realization in optical lattices discussed in Duan, et al. [Phys. Rev. Lett., 91(9):090402, August 2003], incorporated herein by reference. The original microscopics was defined on a honeycomb lattice, but the only essential feature is that the bonds can be grouped into three distinct classes—x, y, and z—so that, at all vertices, exactly one bond from each class is present. This weaker condition allows the construction of such lattices on surfaces of any genus.
Another road to the Ising TQFT is through px+ipy superconducting 2DEGs. Such systems are predicted, based on elementary band theory, to arise in a variety of 2D-systems from a spin-orbit coupled semiconductor with superconductivity imported via proximity effect. Examples include Sau, et al. [arXiv:cond-mat/0907.2239v3], Alicea [arXiv:cond-mat/0912.2115v1], and Qi, et al. [arXiv:cond-mat/1003.5448v1], each of which is incorporated herein by reference. Such systems are chiral topological superconductors and support localized Majorana states.
These systems are not purely topological, but, being superconductors, also support a classical order parameter φ If the system is not planar, but configured as a surface of genus >0, a significant stiffness term λ|▾φ|2 in the Lagrangian may prevent superposition of certain topological states, which are correlated with the winding of φ, a classical quantity for extensive superconductors.
Within the Ising TQFT, braid operations, together with nondemolition measurement of the collective charge of up to four σ-particles, supports the implementation of all Clifford operations. It is known that if the “π/8-gate,”
is added to the Clifford operations, a computationally universal gate results.
Disclosed herein is a protocol that enables the π/8-gate in chiral topological superconductors in which superconducting stiffness λ, has been suppressed. The protocol enables a topologically protected π/8-gate in any pure Ising system that can be fabricated into genus=1 surface. By adding the π/8-gate to previously known techniques, a design for universal topologically protected quantum computation which may be implemented using rather conventional materials may be obtained.
The parent system to a purely Ising TQFT may be any semiconductor/superconducting system resulting in a “generic (no special symmetries) 2D topological superconducting film with a single sheeted Dirac-like Fermi surface.” Our terminology sometimes identifies such a state (called X) in our provisional patent application no. 61/290,645 with the Ising sandwich structure which houses it. A suitable physical substrate for the systems disclosed herein could derive from the experimental effort to realize any of the three “sandwich structures” (see, e.g., Sau, et al., Alicea, and Qi, et al.). Essentially, the stiffness λ may be reduced by (1) quantum fluctuation of φ, e.g., in a granular superconductor where particle number becomes classical and the dual observable φ fluctuates, and (2) a second order phase transition from the px+ipy superconducting state to pure Ising, i.e., induced by condensation of pairs of Abrikosov vortices. The experimental signature of a sandwich structure in a pure Ising state would be (1) the loss of superconductivity, together with (2) the retention of a chiral gapless neutral mode capable of transporting energy at the system edge.
We use the notation X for any 2DEG in a pure Ising TQFT state or, alternatively, superconducting px+ipy 2DEG in which the superconducting stiffness is sufficiently small.
The state Xis housed in a dynamical topology changing device (DTC) and the extent of X at any given instant of time is regulated by electric and/or magnetic gating. A DTC is a 3D device grown in layers using molecular beam epitoxy (MBE) and/or other techniques so that the central 2DEG interface in state Xis contorted into a curved surface of genus=1. Electric and/or magnetic gates can, on a 10−6-10−8 second time scale, regulate exactly which portion of the curved interface is actively in state X at any given time t. The gates do this either by using charge to expel electrons from the 2DEG where they are unwanted, or applying a magnetic field to alter the physical properties of the 2DEG to make it effectively a vacuum and so create a new edge to the state X. The initial qubit is stored in the “pair of points” obtained by expelling state X from the “handle” of the DTC. Since only the topological time sequence of shapes for X is critical, we describe the time history by marking curves and points not included in X(t), X at time=t. Physically, a neighborhood of the indicated curves and/or points will also be “vacuum,” i.e., not in X(t).
lying within the Clifford group. The protocol is a variant of that described in Bonderson, et al. [arXiv:1003.2856v1], incorporated by reference, and its analysis is similar.
The stiffness parameter λ must be zero (or small) because near time t=⅔, as above, the state X occupies a surface of positive genus. From the Ising data table above, the topological charge along the dotted loop γ will be
requiring a superposition of even(2π) winding of φ for |1> and |Ψ> and odd(2π) winding of φ for |σ>.
The central protocol shown in [⅔, 1], the portion of the DTC near the (1, −2)-curve indicated in the leftmost portion of
The above may be refereed to as the “central” protocol. In order to have a complete, functioning protocol for the π/8 gate from a standard fixed planar qubit back to itself, the central protocol may be preceded and followed by teleportation protocols which may be referred to as “into” and “out of” The following diagrams in the notation of MTC describe these teleportation protocols. Note that the “out of” protocol calls for interferometry along a loop δ which, in part, runs through the DTC in the rightmost depiction in
Anyonic teleportation of quantum information in and out of twisted qubits is shown in
The composition of protocols “into,” “central,” and “out of” realizes in a topologically protected manner either the π/8 gate or its inverse. The measurement outcome |1> or |Ψ> during the central protocol tells us which. If |Ψ> is observed, the inverse gate is effected. To rectify that situation, composition with the braid generator
is required. We call the totality of steps the “final protocol.”
Numerous other general purpose or special purpose computing system environments or configurations may be used. Examples of well known computing systems, environments, and/or configurations that may be suitable for use include, but are not limited to, personal computers, server computers, hand-held or laptop devices, multiprocessor systems, microprocessor-based systems, set top boxes, programmable consumer electronics, network PCs, minicomputers, mainframe computers, embedded systems, distributed computing environments that include any of the above systems or devices, and the like.
Computer-executable instructions, such as program modules, being executed by a computer may be used. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Distributed computing environments may be used where tasks are performed by remote processing devices that are linked through a communications network or other data transmission medium. In a distributed computing environment, program modules and other data may be located in both local and remote computer storage media including memory storage devices.
With reference to
Computer 110 typically includes a variety of computer readable media. Computer readable media can be any available media that can be accessed by computer 110 and includes both volatile and nonvolatile media, removable and non-removable media. By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. Computer storage media includes both volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CDROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can accessed by computer 110. Communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, infrared and other wireless media. Combinations of any of the above should also be included within the scope of computer readable media.
The system memory 130 includes computer storage media in the form of volatile and/or nonvolatile memory such as read only memory (ROM) 131 and random access memory (RAM) 132. A basic input/output system 133 (BIOS), containing the basic routines that help to transfer information between elements within computer 110, such as during start-up, is typically stored in ROM 131. RAM 132 typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by processing unit 120. By way of example, and not limitation,
The computer 110 may also include other removable/non-removable, volatile/nonvolatile computer storage media. By way of example only,
The drives and their associated computer storage media discussed above and illustrated in
The computer 110 may operate in a networked environment using logical connections to one or more remote computers, such as a remote computer 180. The remote computer 180 may be a personal computer, a server, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to the computer 110, although only a memory storage device 181 has been illustrated in
When used in a LAN networking environment, the computer 110 is connected to the LAN 171 through a network interface or adapter 170. When used in a WAN networking environment, the computer 110 typically includes a modem 172 or other means for establishing communications over the WAN 173, such as the Internet. The modem 172, which may be internal or external, may be connected to the system bus 121 via the user input interface 160, or other appropriate mechanism. In a networked environment, program modules depicted relative to the computer 110, or portions thereof, may be stored in the remote memory storage device. By way of example, and not limitation,
| Number | Date | Country | |
|---|---|---|---|
| 61290661 | Dec 2009 | US | |
| 61290645 | Dec 2009 | US |
