Robust Majorana magic gate via measurement

Abstract

Embodiments of the disclosed technology relate to a method for implementing a π / 8 phase gate in a quantum computing device. In certain embodiments, a quantum circuit is evolved from an initial state to a target state using a hybrid measurement scheme. The hybrid measurement scheme can include applying one or more measurements to the quantum circuit that project the quantum circuit towards the target state; and applying one or more adiabatic or non-adiabatic techniques that adiabatically evolve the quantum circuit towards the target state.

Description

Background Art

[0001] This application relates to quantum computing devices, and more particularly, to π / 8 phase gates implemented in a topologically protected system. Summary of the Invention

[0002] The π / 8 phase gate (magic gate) is an ideal component for enhancing topological systems based on Majorana zero modes to achieve full quantum universality. Example embodiments of a scheme based on a combination of projective measurements and quantum evolution (e.g., non-adiabatic evolution) are disclosed herein, where the quantum evolution effectively eliminates smooth control errors when implementing phase gates in Majorana-based systems. Previous schemes based on adiabatic evolution are vulnerable to problems caused by small and finite dynamic phases typically present in topologically unprotected gates. Measuring alone eliminates the dynamic phase. However, for unprotected gates, the forced measurement scheme is no longer effective, resulting in a low success probability of obtaining correct consecutive measurement results in a measurement-only implementation. In the present disclosure, it is shown how to obtain a viable measurement-based scheme that significantly increases the success probability by evolving the system non-adiabatically with respect to the subspace of phase shifts between measurements.

[0003] In some embodiments, the quantum state of a quantum circuit configured to implement a π / 8 phase gate in a quantum computing device is changed from an initial state to a target state using a hybrid measurement scheme. In the illustrated embodiment, the hybrid measurement scheme includes: applying one or more measurements to the quantum state that project the quantum state towards the target state; and applying one or more adiabatic or non-adiabatic techniques that evolve the quantum state towards the target state.

[0004] In certain embodiments, the quantum computing device is a topologically protected quantum computing device. In some embodiments, the hybrid measurement scheme reduces timing noise, slow parameter noise, dynamic phase noise, and / or parallel dissipation. In certain embodiments, one or more measurements of the quantum state are applied between applications of any of the following: (a) two non-adiabatic techniques in a non-adiabatic technique; or (b) two adiabatic techniques in an adiabatic technique. In some embodiments, the application of one or more measurements of the quantum state includes general geometric decoupling of the quantum circuit. For example, general geometric decoupling can be performed by applying a continuous projection operator to the quantum circuit. In some examples, the continuous projection operator is applied at turning points of the geometric decoupling trajectory, and the turning points are determined using Chebyshev polynomials. In further examples, the general geometric decoupling of the quantum circuit maps to multiple parameter scans across the poles of the unit sphere.

[0005] Any of the embodiments disclosed above may be implemented as part of a system that includes: a quantum computing device that includes a quantum circuit; and a classical computing device that communicates with the quantum computing device and is adapted to perform any of the disclosed methods. Any of the embodiments disclosed above may also be implemented by one or more computer-readable media storing computer-executable instructions that, when executed by a classical computer, cause the classical computer to perform a method of controlling a quantum computing device in accordance with any of the disclosed methods.

[0006] The foregoing and other objects, features, and advantages of the disclosed technology will become more apparent from the following detailed description taken in conjunction with the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0007] FIG. 1(a) is a schematic block diagram illustrating a visualization of a swap process of lines that are octants covering a unit sphere.

[0008] FIG. 1(b) is a schematic block diagram illustrating a visualization of a sequence of π / 8 gates in an ideal Y-junction system.

[0009] Figure 2 is a schematic block diagram illustrating an evolution-based geometric decoupling scheme.

[0010] FIGS. 3(a) and 3(b) are schematic block diagrams illustrating only a geometric decoupling scheme.

[0011] Figure 4 is a schematic block diagram illustrating a visualization of an example hybrid protocol.

[0012] Figure 5 Shows a number of diagrams that illustrate simulation results that illustrate aspects of the disclosed technology.

[0013] Figure 6 Illustrates an example generalization of a suitable classical computing environment in which several of the described embodiments may be implemented.

[0014] Figure 7 Illustrates an example of a possible network topology (e.g., a client-server network) for implementing a system according to the disclosed technology.

[0015] Figure 8 Illustrates another example of a possible network topology (e.g., a distributed computing environment) for implementing a system according to the disclosed technology.

[0016] Figure 9 Illustrates an exemplary system for implementing the disclosed technology.

[0017] Figure 10 It is a flowchart showing an exemplary embodiment that generalizes an embodiment for implementing the disclosed technology. Detailed Description

[0018] I. General Considerations

[0019] As used in this application, the singular forms "a", "an", and "the" include plural forms unless the context clearly dictates otherwise. Additionally, the term "includes" means "comprises". Further, the term "coupled" does not exclude the presence of intermediate elements between the coupled items. Further, as used herein, the term "and / or" refers to any one or any combination of the items in the phrase.

[0020] Although the operations of some of the disclosed methods are described in a specific, sequential order for convenient presentation, it should be understood that this description includes rearrangements unless the specific language set forth below requires a particular order. For example, operations described sequentially may in some cases be rearranged or performed concurrently. Additionally, for simplicity, the figures may not show the various ways in which the disclosed systems, methods, and devices may be used in conjunction with other systems, methods, and devices. Additionally, the description sometimes uses terms such as "generate" and "provide" to describe the disclosed methods. These terms are high-level abstractions of the actual operations being performed. The actual operations corresponding to these terms will vary depending on the particular implementation and will be readily discernible to any person of ordinary skill in the art.

[0021] II. Overview

[0022] The embodiments disclosed herein are a new way to implement a π / 8 gate (also known as the magic gate or T gate) for quantum computing, which eliminates the problems of common noise sources that limit other ways. Embodiments of the scheme disclosed herein employ a combination of measurement and quantum evolution, which is hereinafter referred to as the hybrid measurement scheme (HMS). High-fidelity π / 8 gates are a valuable component for most universal quantum computing proposals. The HMS will be particularly important for universal topological quantum computing. Current leading ways use Majorana zero modes (MZMs) to implement high-fidelity Clifford gates, but need to be augmented by π / 8 gates. Although a process called magic state distillation can reduce the error of the π / 8 gate to any amount, it requires a large physical qubit overhead and trial π / 8 states (produced by noisy π / 8 gates). See, for example, S. Bravyi and A. Y. Kitaev, Phys. Rev. A 71, 022316 (2005), quant-ph / 0403025. Providing high-quality trial π / 8 states for the distillation process significantly reduces the overhead and will be of significant practical importance for future implementations of topological quantum computing. By eliminating the main noise sources of the π / 8 gate in a Majorana-based quantum computing architecture, the HMS can be expected to significantly improve the quality of trial π / 8 states compared to all other ways known today.

[0023] A. Noise sources

[0024] The following noise sources can be systematically eliminated by example embodiments of the HMS.

[0025] Timing noise. The finite energy splitting of qubits causes the accumulation of dynamic phase differences over time. Most proposals for the π / 8 gate rely on the fine-tuning of the accumulated dynamic phase within certain time intervals, and thus require precise control of the time and amplitude of the applied energy splitting.

[0026] Calibration errors. Calibration errors are systematic and unknown offsets of the control parameters of the system from their expected values.

[0027] Slow parameter noise. Even if calibration errors can be avoided by extensive benchmarking and readjustment of the system, time-dependent changes in some of the system parameters, even if slow, result in new calibration errors.

[0028] Residual dynamic phase noise. Initial approaches sought to eliminate the three noise sources described above. See, e.g., T. Karzig, Y. Oreg, G. Refael, and M. H. Freedman, Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161. The corresponding scheme is based on adiabatically evolving the system and is named universal geometric decoupling. The limitation of this approach is that for Majorana-based systems, it is generally not possible to keep the qubit energy splitting zero throughout the evolution implementing the π / 8 gate. This results in a small residual dynamic phase. This initial approach aimed to use an echo process to eliminate the residual dynamic phase noise. However, timing noise and dissipation will limit the effectiveness of the echo.

[0029] Parallel dissipation. Here, environmental noise that changes the Hamiltonian and commutes with the original Hamiltonian of the system is represented as parallel dissipation. This type of noise will cause a phase shift of the energy eigenstates of the system but will not cause transitions between different eigenstates. In a timing-based approach, parallel dissipation will cause a phase shift of the qubit. In T. Karzig, Y. Oreg, G. Refael, and M. H. Freedman, Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161, due to the residual dynamic phase, parallel dissipation will cause a phase shift. Additionally, parallel dissipation is predicted to increase the time scale required to reach the adiabatic regime (see, e.g., C. Knapp, M. Zaletel, D. E. Liu, M. Cheng, P. Bonderson, and C. Nayak, Phys. Rev. X 6, 041003 (2016), arXiv:1601.05790), which will require more time to implement the scheme (see, e.g., T. Karzig, Y. Oreg, G. Refael, and M. H. Freedman, Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161).

[0030] To date, no known non-topological or Majorana-based scheme can systematically eliminate all of the above noise sources. Topological phases beyond MZMs can in principle avoid the above noise sources. However, to date, the corresponding topological phases have not been experimentally realized with sufficient control to be useful for quantum computing.

[0031] B. Overview of HMS

[0032] The details of HMS will be discussed in the following sections. Here, some concepts of HMS are described.

[0033] The HMS uses general geometric decoupling to eliminate timing noise, calibration errors, and slow parametric noise. General geometric decoupling can be considered a particularly robust alternative to repeated parametric sweeps aimed at eliminating the above errors. Instead of the adiabatic evolution of the underlying quantum state, the HMS uses measurements that project the system onto special points in the parameter space (e.g., turning points of the evolution, as identified in Karzig, Y. Oreg, G. Refael, and M. H. Freedman, Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161n Ref.). This allows avoiding certain intermediate points where the system parameters are unfavorable and would otherwise lead to residual dynamic phase noise.

[0034] Performing the measurement does not deterministically project onto a specific state. In particular, the proposed measurement typically has two measurement outcomes, one of which is the outcome that confirms the desired projection, and the other outcome indicates projection onto a different (undesired) state. Obtaining the undesired projection is taken as a failure of the π / 8 state preparation process, which will then be restarted. To increase the probability of a successful HMS, there is non-adiabatic Hamiltonian evolution applied to the system between measurements, which rotates the state towards the target state of the measurement projection. This greatly increases the chance of obtaining the desired measurement outcome. Small errors in the application of this rotation (either through timing noise in the non-adiabatic evolution or by changing the strength of the Hamiltonian through parallel dissipation) only result in a small reduction in the probability of performing the desired projection. In particular, obtaining the desired projection outcome will result in a state that is not affected by errors in the non-adiabatic Hamiltonian evolution. As a result, errors will generally only reduce the success probability of the HMS. However, if the HMS is successful, the resulting π / 8 state will be of high quality. In the following section, a discussion is provided that shows that even for relatively large errors in the applied non-adiabatic evolution, the success probability of the HMS is still large enough (e.g., on the order of 50%) for a practically relevant cancellation protocol for removing the large number of noise sources mentioned in Section IIA.

[0035] The combination of projection measurements that ensure high fidelity of the resulting state when the desired measurement outcome is obtained and non-adiabatic evolution that increases the chance of obtaining the desired measurement outcome is unique to the disclosed HMS embodiments. Using this process, high-quality π / 8 states can be prepared. Compared to other existing schemes, embodiments of the HMS scheme can eliminate the error sources listed in Section IIA. Therefore, this is an ideal approach for quantum computing platforms using π / 8 gates.

[0036] III. Background and Further Introduction

[0037] Topological quantum computation holds the promise for inherently error-protected storage devices and quantum information operations using the braiding of non-Abelian anyons. See, for example, A.Y. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics 303, 2 (2003), quant-ph / 9707021. Majorana zero-energy modes (MZMs) form the simplest non-Abelian anyons. The formation of such states in one and two dimensions is theoretically predicted to occur in quantum Hall states and certain semiconductor-superconductor devices.See, for example, N. Read and D. Green, “Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect,” Phys. Rev. B 61, 10267 (2000), cond-mat / 9906453; A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires,” Physics Uspekhi 44, 131 (2001), cond-mat / 0010440; J. D. Sau, S. Tewari, R. M. Lutchyn, T. D. Stanescu, and S. Das Sarma, “Non-Abelian quantum order in spin-orbit-coupled semiconductors: Search for topological Majorana particles in solid-state systems,” Phys. Rev. B 82, 214509 (2010), arXiv:1006.2829; R. M. Lutchyn, J. D. Sau, and S. Das Sarma, “Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures,” Phys. Rev. Lett. 105, 077001 (2010), arXiv:1002.4033; Y. Oreg, G. Refael, and F. von Oppen, “Helical Liquids and Majorana Bound States in Quantum Wires,” Phys. Rev. Lett. 105, 177002 (2010), arXiv:1003.1145。In the past decade, Majorana modes have indeed appeared in the reports of multiple experiments (for a review, see R.M. Lutchyn, E.P.A.M. Bakkers, L.P. Kouwenhoven, P. Krogstrup, C.M. Marcus, and Y. Oreg, “Realizing Majorana zeromodes in superconductor–semiconductor heterostructures,” Nat. Rev. Mater. (2018), arXiv:1707.04899), which has raised the prospects for Majorana-based topological quantum computers.

[0038] However, MZMs are not complex enough to allow for a dense packing of the Hilbert space of computation and thus cannot perform universal topological quantum computation. See, for example, S. Bravyi and A.Y. Kitaev, “Universal quantum computation with ideal clifford gates and noisy ancillas,” Phys. Rev. A 71, 022316 (2005), quant-ph / 0403025. Although the braiding of MZMs can perform topologically protected Clifford gates, they cannot implement topologically protected magic gates or generate magic states (also known as T gates or π / 8 phase gates), which are necessary to complete Clifford gates for universal quantum computation.

[0039] There are proposals to enhance Majorana-based architectures with magic gates. However, these proposals are typically unprotected and range from exact timing to fine-tuning geometry approaches. See, for example, J.D. Sau, S. Tewari, and S. Das Sarma, “Universal quantum computation in a semiconductor quantum wire network,” Phys. Rev. A 82, 052322 (2010), arXiv:1007.4204; F. Hassler, A.R. Akhmerov, and C.W.J. Beenakker, “The top-transmon: a hybrid superconducting qubit for parity-protected quantum computation,” New J. Phys. 13, 095004 (2011), arXiv:1105.0315; T. Hyart, B. van Heck, I.C. Fulga, M. Burrello, A.R. Akhmerov, and C.W.J. Beenakker, “Flux-controlled quantum computation with Majorana fermions,” Phys. Rev. B 88, 035121 (2013), arXiv:1303.4379; D.J. Clarke, J.D. Sau, and S.D. Das Sarma, “A Practical Phase Gate for Producing Bell Violations in Majorana Wires,” Phys. Rev. X 6, 021005 (2016), arXiv:1510.00007; S. Plugge, L.A. Landau, E. Sela, A. Altland, K. Flensberg, and R. Egger, “Roadmap to Majorana surface codes,” Phys. Rev. B 94, 174514 (2016), arXiv:1606.08408; S. Plugge, A. Rasmussen, R. Egger, and K. Flensberg, “Majorana box qubits,” New J. Phys. 19, 012001 (2017), arXiv:1609.01697. The exception is the proposal to develop highly specialized genon-based hardware to produce topologically protected magic gates.See, for example, M. Barkeshli, C.-M. Jian, and X.-L. Qi, “Twist defects and projective non-abelian braiding statistics,” Phys. Rev. B 87, 045130 (2013), arXiv:1208.4834; M. Barkeshli and J. D. Sau, “Physical Architecture for a Universal Topological Quantum Computer based on a Network of Majorana Nanowires,” (2015), arXiv:1509.07135. In contrast, the current authors have proposed geometric protocols that are robust against systematic errors. See, for example, T. Karzig, Y. Oreg, G. Refael, and M. H. Freedman, “Universal Geometric Path to a Robust Majorana Magic Gate,” Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161. From a geometric perspective, the π / 4 phase gate corresponding to the exchange of two MZMs corresponds to a topologically protected adiabatic path that encircles an octant in the Bloch sphere of the parameter space, see Figs. 1(a) and 1(b). By traversing the geometric space in an alternating manner, a geometric decoupling scheme for the π / 8 phase can be implemented that effectively eliminates systematic errors, such that the remaining errors are exponentially small for a correctly chosen number of turns at the zeros of the Chebyshev polynomials. T. Karzig, Y. Oreg, G. Refael, and M. H. Freedman, “Universal Geometric Path to a Robust Majorana Magic Gate,” Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161. As reviewed in Section IVA, however, unfortunately and unavoidably, away from the protected path defined by the edge of the octant, the π / 8 gate cannot be implemented in a fully geometric way because typically the system will pick up a small but finite dynamical phase that must be eliminated by conventional error correction protocols (e.g., via an echo sequence).

[0040] IV. Example Embodiments of the Disclosed Technology

[0041] In this disclosure, and in certain example embodiments, only the elements of measurement-based topological quantum computation are used to overcome these dynamical errors. See, e.g., P. Bonderson, M. Freedman, and C. Nayak, “Measurement-only topological quantum computation,” Phys. Rev. Lett. 101, 010501 (2008), arXiv:0802.0279. In this disclosure, a series of protocols are demonstrated, starting from a simple measurement-based implementation of the geometric magic gate of T. Karzig, Y. Oreg, G. Refael, and M. H. Freedman, “Universal Geometric Path to a Robust Majorana Magic Gate,” Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161, then modifying the sequence by additional measurements, and finally, combining dynamical evolution and measurement to yield a superior protocol that systematically eliminates all major error sources.

[0042] In general, one might expect that a measurement-only scheme would perform better than an adiabatic braiding approach because a part of the information (the result of a projective measurement) is classically stored and used and thus does not undergo any decoherence effects of quantum computation. To illustrate this, consider four MZM operators (γ0, γ x , γ y , γ z ). Using the relation {γi ， γ j} = 2δ ij , i, j = 0, x, y, z. With the help of the ancillas γ0, γ z , it can be easily checked that the braiding operators γ given by x and γ y can be implemented by a series of projections P z P y P x P z = BxyP z / √8, where P i = 1 - iγ0γ i ) / 2 is the projective measurement operator for the unoccupied state of the MZM pair γ0 and γ i . Since in a typical measurement the probability of measuring a pair of MZM parity checks equals 1 / 2, e.g., projecting the pair to the unoccupied state, the total probability of applying the above projections equals 1 / 2 3= 1 / 8. Typically, obtaining other measurement results creates different gates, which may require appropriate corrections depending on the results. An alternative is to use a forced measurement scheme, where a pair of measurement processes is repeated until the desired measurement result is obtained. See, for example, P. Bonderson, M. Freedman, and C. Nayak, “Measurement-only topological quantum computation,” Phys. Rev. Lett. 101, 010501 (2008), arXiv:0802.0279.

[0043] The structure of the present disclosure is as follows. In Section IVB1, the modification from the adiabatic geometric decoupling scheme to a measurement-only process is described. The application of the continuous projection operator at the turning points of the geometric decoupling trajectory of the adiabatic scheme is discussed. The measurement-only scheme allows avoiding the case where all Majorana couplings are significant, i.e., the case where the dynamic phase is accumulated. While this eliminates the need for echo error correction, the success probability in this scheme becomes state-dependent for the qubits and results in a small deviation from the desired phase gate. In Section IVB2, it is shown how these deviations can be avoided by a forced measurement echo process, reminiscent of the dynamic phase elimination echo of the adiabatic scheme. In Section IVB3, the north / south projection protocol is discussed, which eliminates the need for the echo process and renders an exact magic gate, but the success probability drops to 2 -N , where N is the number of steps in the geometric decoupling protocol. Since only successful results are fed into the subsequent distillation scheme, a small success probability is not a problem in principle. However, increasing the success probability will drastically reduce the time to prepare the magic state.

[0044] Embodiments of the disclosed technology provide protocols that combine the dynamic evolution with the measurement steps. In Section IVC, a novel hybrid evolution / measurement approach is shown that increases the success probability to O(1), while also producing an exact high-quality phase gate.

[0045] Section IVD is dedicated to the numerical implementation of the hybrid evolution, which exemplifies the various methods discussed in Sections IVB and IVC. In Section IVE, the invention content is provided.

[0046] A. Review of Geometric Decoupling

[0047] The main problem in implementing a robust magic gate is the extreme fine-tuning required of the qubit Hamiltonian. Despite their topological protection, MZMs are no exception. However, MZMs have a relative advantage over non-topological qubits because it is possible to utilize the geometric phase. Below, the process of obtaining magic states using MZMs is reviewed, including the main drawbacks of the process and how to use a universal geometric decoupling process to overcome most of the errors. See, for example, T. Karzig, Y. Oreg, G. Refael, and M. H. Freedman, “Universal Geometric Path to a Robust Majorana Magic Gate,” Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161.

[0048] The geometric path to the magic gate is best illustrated by a Y-junction system. Three Majorana modes, γ x 、γ y and γ z , are located at the tips of the Y-junction and interact only with a fourth MZM, γ0, which is at the center of the junction and has a Hamiltonian:

[0049]

[0050] where the following definitions are applied: the Majorana vector γ = (γ x , γ y , γ z ) and the coupling unit vector h = (h x , h y , h z ). The Y-junction coupling, ωh i, depends exponentially on physical parameters such as the distance between MZMs or the gating confinement potential (see, e.g., B. van Heck, A. R. Akhmerov, F. Hassler, M. Burrello, and C. W. J. Beenakker, “Coulomb-assisted braiding of Majorana fermions in a Josephson junction array,” New J. Phys. 14, 035019 (2012), arXiv:1111.6001; T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freedman, “Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes,” Phys. Rev. B 95, 235305 (2017), arXiv:1610.05289). This motivates the fundamental assumption in this paper that these couplings can be adjusted such that their ratio reaches 0 or ∞ with exponential precision.

[0051] 1. Exchange process and its π / 8 (magic) generalization

[0052] The exchange process of MZMs in this system can be achieved by adjusting the coupling strength ωh i starting from h z ≈1 >> h x , h y .γ x and γ y being the zero modes of the problem. To exchange them, move h x ≈1 >> h z , h y in a continuous manner while keeping h y << 1. Followed by h y ≈1 >> h x , h z (while keeping h z << 1), and finally restore the system to its original state h z >> h x , h y (while keeping h x << 1).

[0053] Such an operation can be geometrically visualized. Let us consider h as a 3D vector and represent it in spherical coordinates. See, for example, C.-K. Chiu, M. M. Vazifeh, and M. Franz, “Majorana fermion exchange in strictly one-dimensional structures,” EPL 110, 10001 (2015), arXiv:1403.0033. h is the radius vector, using the polar angle and the azimuthal angle θ and and their unit vectors e θ and In this way, the Majorana is represented as:

[0054]

[0055] is a zero mode that commutes with the Hamiltonian (1). The exchange process can now be easily visualized as h marking the octant of the unit sphere, with the boundary at θ = π / 2 and planes (see Fig. 1(a)).

[0056] The effect of this adiabatic operation on the two zero-mode states is encapsulated in the Berry phase of the Bloch sphere partitioned by h. Representing a single fermionic annihilation operator from the two zero modes as:

[0057]

[0058] This operator connects two parity states, |0> (defined as a|0> = 0, and ). In response to an adiabatic closed operation on the vector h, these states change to:

[0059] U c |(1 ± 1) / 2> = e ±1α |(1 ± 1) / 2> (4)

[0060] where the phase difference 2α is given by the solid angle enclosed by the delineated contour. For the octant, α exchage = π / 4.

[0061] Obtaining the magic π / 8 gate now seems obvious. All that is needed is to cover half of the solid angle covered by the exchange process. For example, one can go from changing θ = 0 → π / 2, then and return θ = π / 2 → 0, and finally, closing the trajectory (see Fig. 1(b)).

[0062] More specifically, FIG. 1(a) is a schematic block diagram 100 showing the visualization of the swapping process of the lines that cover the octants of the unit sphere. The lines start from the north pole (h z >>h x , h y ), then continue to point X on the equator (h x >>h y , h z ), then point Y (h y >>h x , h z ), and finally reach the north pole again (h z >>h x , h y ), thus completing the cycle. The Berry phase difference of the two parity sectors accumulated during this process is equal to half of the covered solid angle, π / 2.

[0063] FIG. 1(b) is a schematic block diagram 110 showing the visualization of the sequence for the π / 8 gate in an ideal Y-junction system. This trajectory is not protected because while modifying h z , h x = h y must be maintained, and small fluctuations will produce different phases.

[0064] Although the geometric magic gate is elegant, it also suffers from obvious drawbacks. The plane is a fine-tuning band of the parameter space, which requires maintaining h x = h y . However, such control is unrealistic, and control errors lead to arbitrary errors in the calculation. Additionally, the π / 8 trajectory ideally passes through the region where all three Majorana couplings have similar strengths This will inevitably cause the next-nearest-neighbor coupling between the Majorana modes at the top of the Y-junction, which will split the ground-state degeneracy between the two parity states and cause an arbitrary dynamic phase state between the |0> and |1> states. Below, the direct couplings between the MZMs, γ x , γ y , and γ z will be referred to as the external couplings.

[0065] 2. General Geometric Decoupling

[0066] The key in T. Karzig, Y. Oreg, G. Refael, and M. H. Freedman, “Universal Geometric Path to a Robust Majorana Magic Gate,” Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161 is to indicate that by using iterative and universal trajectories through the h-sphere, the above system control errors can be eliminated to arbitrary precision. The idea comes from the intuition of the snake-like trajectories as in Figure 2 which can essentially average out the errors due to faulty device control. The turning points n = 1,..., 2N can be optimized to systematically eliminate the errors in the accumulated phase, recalling the concept of universal dynamical decoupling. (See, e.g., G. S. Uhrig, “Keeping a quantum bit alive by optimized pulse sequences,” Phys. Rev. Lett. 98, 100504 (2007), quant-ph / 0609203.). More specifically, Figure 2 is a schematic block diagram 200 illustrating an evolution-based geometric decoupling scheme. The correct choice of the turning points yields a trajectory that covers a solid angle of π / 4 with exponentially smaller errors. Here, for the Chebyshev polynomial with N = 5, the contours are plotted, and n = 1,..., 2N are given in Eq. (6).

[0067] Specifically, as long as the errors at the turning points are systematic and are described by a smooth function the turn number δα ∼ e -2N in the gate error can be exponentially suppressed. Then the optimal turning points can be derived by expanding the error in terms of Chebyshev polynomials and eliminating the first 2N - 1 orders of the expansion. See T. Karzig, Y. Oreg, G. Refael, and M. H. Freedman, “Universal Geometric Path to a Robust Majorana Magic Gate,” Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161. This process yields 2N equations

[0068]

[0069] where m = 1... 2N, where T m * (x) = Tm (2x - 1) is the first-kind shifted Chebyshev polynomial. For α = π / 8, the magic gate is implemented. In this case, the solution can be expressed analytically and is given by

[0070]

[0071] The Chebyshev protocol, while effectively eliminating systematic machine errors, does not solve the problem of uncontrollable dynamic phases caused by finite external couplings when all couplings h i are strong. In T. Karzig, Y. Oreg, G. Refael, and M. H. Freedman, “Universal Geometric Path to a Robust Majorana Magic Gate,” Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161, it is shown how this dynamic phase can be eliminated by performing an echo sequence. However, echo sequences can prove costly as they extend the computation time and strongly depend on the stability of the system. Many current works attempt to completely eliminate the need for echoes by avoiding regions where all three couplings ωh i are large. This can be done using measurement-based methods, as shown below.

[0072] B. Measurement-only approach

[0073] In the measurement-only approach, the adiabatic evolution of the state is replaced by a set of measurements. (See, for example, P. Bonderson, M. Freedman, and C. Nayak, “Measurement-only topological quantum computation,” Phys. Rev. Lett. 101, 010501 (2008), arXiv:0802.0279). Here, the focus is on determining measurements of the (joint) parity of a set of MZMs. With knowledge of the measurement results, the measurement results can be described by the projection P p or P p - where p represents the parity of the chosen set of MZMs, and P p (P p - ) can be defined as the projection onto p = 1 (p = -1). A series of measurements then acts on the initial state |ψ>, resulting in the (normalized) final state p s -1 / 2 Π j P jThe product of the projections of |ψ>, where p s represents the probability of obtaining a particular set of measurement results.

[0074] 1. Direct conversion of evolution-based to measurement-only geometric decoupling

[0075] Figures 3(a) and 3(b) are schematic block diagrams 300 and 310 illustrating the measurement-only geometric decoupling scheme. The projection operators are applied in the order indicated below each panel. Figure 3(a) illustrates the Figure 2 direct conversion of the evolution-based scheme in

[0076] to a measurement-only implementation. Figure 3(b) illustrates the measurement-only implementation of the north / south scan protocol. along the equator to and then back to the north pole is described by a set of projections:

[0077]

[0078] where

[0079]

[0080] 2P φ = 1 - iγ0[cos(φ)γ x + sin(φ)γ y (8)

[0082] In the following, it is convenient to rewrite the MZMs in terms of the Pauli matrices σ j as iγ0γ x = σ x , iγ0γ y = σ y and -iγ x γ y = σ z Note that iγ0γ z = σ z τ z , where the Pauli matrix τ z = -γ x γ y γ z γ0 describes the overall parity of the four MZM systems. For concreteness, consider the case of a system with 6 MZMs and with a fixed (even) parity, where one qubit is encoded in 4 MZMs γ x ​, γ y , γ1, γ2, where γ0, γ z is used as an auxiliary. In this case, τ z = iγ1γ2 describes the z - Pauli operator of the qubit.

[0083] In this notation, it is found that

[0084]

[0085] where P zτ projects onto the space where σ z = -τ z The formula (10) allows the effects of the projection set to be divided into three different contributions: (1) The projection P zτ fixes the state of the degrees of freedom of the auxiliary (σ). (2) The unitary operator acts as a phase gate on the degrees of freedom of the qubit (τ). The origin of the phase gate comes from the different geometric phases accrued when completing a cyclic path in the Bloch sphere of the auxiliary (σ). Depending on whether τ z = +1 (-1), the path starts and ends at the north (south) pole. The opposite geometric phases in the two cases then act as a phase gate. (3) The pre - factor describes the success probability of obtaining the measurement result

[0086] Note that the phase gate implemented by is the same as the adiabatic evolution along the geodesic connecting the projection points on the Bloch sphere. This agreement allows the implementation of the same geometric decoupling scheme as in Section IVA in a measurement - only setting. The complete π / 8 gate will be implemented by However, there is an important difference: only a specific set of measurement results will produce a single projection

[0087] Now, the present disclosure studies the result operations for different measurement results. Flipping has a relatively small effect. If both angles are shifted by π, the same gate is still obtained. If only one angle is shifted, the gate will be different from the entire τ z gate, which can be easily recorded and (if necessary) corrected. If the z - projection is shifted P zτ → P z-τ , then problems will occur. The latter will lead to a random sign flip To avoid this problem, forced measurements can be used (see, e.g., P. Bonderson, M. Freedman, and C. Nayak, “Measurement-only topological quantum computation,” Phys. Rev. Lett. 101, 010501 (2008), arXiv:0802.0279) by repeating measurements along the and Z τ axes until z τ = +1 is found. Note that the different paths in this correction process only result in a phase difference that is a multiple of 2π, which can be ignored (e.g., compare the geometric phase of the path through the point where Thus, the forced measurement procedure increases the success probability of the entire measurement protocol from 2 -N to unity.

[0088] 2. Role of the external coupling

[0089] One of the motivations for using a measurement-based scheme to achieve geometric decoupling is that by using projections instead of adiabatic time evolution, the intermediate region of the octant where all ωh i Majorana couplings are turned on can be avoided. As mentioned in Section IVA, the danger of this mechanism is that the inevitable second-order coupling between "external" MZMs (e.g., between γ x and γ y ) leads to the splitting of the qubit state degeneracy. The accompanying dynamic phase must then be eliminated by an additional echo process.

[0090] When applying projections of the type of formula (10), the dynamic phase does not appear because there is always at least one MZM that is not touched, thus ensuring perfect ground state degeneracy. However, by applying a modified projection alternative the external coupling can lead to different error sources

[0091]

[0092] . This form of projection appears when considering how to physically implement the corresponding measurement. The measurement of the MZM parity can be considered a two-step process. First, the four-fold ground state degeneracy is split (once) by introducing a coupling between the MZMs, either internally or by coupling to a measurement device. This process is described by the Hamiltonian H M . The finite energy splitting then allows the measurement device to determine whether the system is in the ground state or an excited state through the appropriate energy spectrum. Below, the measurement involving turning on the Hamiltonian H M and then projecting onto the corresponding energy eigenstate is described.

[0093] The measurement of the form (11) is implemented via the Hamiltonian:

[0094]

[0095] where quantifies the ratio of the external coupling to the internal coupling. Although in principle it can be fine-tuned to In general, one expects the second-order coupling where Δ0 is a higher energy scale (e.g., the topological gap), which is integrated out to obtain the effective MZM Hamiltonian H M .

[0096] finite The effect is no longer to project directly onto the equator of the Bloch sphere in (σ x , σ y , σ z τz ) , but to points that are slightly shifted north or south. Interestingly, due to the different states of the τ z qubit, these changes are opposite and they do not affect the geometric phase accrued during the application . In particular, for the linear order in , one can find (up to an overall phase)

[0097]

[0098] where Although the finite does not change the applied phase rotation, the (real) prefactor now becomes dependent on τ z . The latter follows intuitively from the different success probabilities of the projection, since depending on the state of the τ z qubit, the projection is either close to the north pole or far from the north pole. Unfortunately, due to the presence of the prefactor that depends on τ z , projections of the form of formula (13) can no longer be used to prepare exact magic states. Since the X eigenstate (|0> + |1>) / √2 can be prepared with topological precision, an exact method for preparing magic states is to apply a π / 8 phase gate to the initial X state. Applying a gate of the form will rotate the X state out of the equator and introduce errors.

[0099] Note that additional echoes similar to canceling the dynamic phase in evolution-based schemes can be used to cancel the prefactor that depends on τ z : First, apply a geometric decoupling protocol to implement a π / 16 phase gate, which includes some unwanted overall prefactor Then, use a forced measurement scheme to project south instead of north and reverse the order of the turning points to still implement the π / 16 gate (this step is equivalent to flipping the qubit and applying a -π / 16 gate). The result will be a π / 8 gate, where the τ z dependence in the prefactor is eliminated.

[0100] 3. South / North Projection Protocol

[0101] In the protocols discussed in Sections IVB1 and IVB2, the evolution and measurement-only approaches are very similar, essentially a one-to-one mapping of their advantages (eliminating systematic errors) and disadvantages (requiring some echo procedure). Now, a different protocol is proposed that allows the measurement-only scheme to eliminate the effects of unwanted external couplings without additional echoes.

[0102] The minimal building block of the protocol is given by and describes the projection of the Bloch sphere from north to south and back (see Fig. 3(b)). Since any projection is surrounded by a projection with the opposite sign in the Z direction, only terms that do not commute with σ z τ z survive, which eliminates the unwanted term iγ x γ y = -σ z . The resulting projection yields

[0103]

[0104] where the implemented gate has the same form as the version in Eq. (10), except that the accumulated phase is now doubled. Thus, using consecutive projections at the turning points that are suitable for the π / 16 gate in the original protocol (e.g., the solution of Eq. (5) for α = π / 16) achieves a complete geometric decoupling scheme. Conceptually, a similar process is possible in adiabatic evolution-based schemes. However, the requirement to control the Hamiltonian to achieve the desired dynamic phase cancellation is much more difficult. One needs to change the sign of the Z component of the Hamiltonian while keeping the X and Y parts exactly the same as the evolution through the northern hemisphere. For the measurement-only version, projecting to the north or south pole corresponds to applying exactly the same measurement; one can simply choose different measurement outcomes. One can simply choose for different measurement outcomes. From Eq. (15), it can be seen that the finite angle

[0105] ​has little impact as it only slightly reduces the success probability of a set of measurements. Similar to Section IVB1, this protocol applies to any measurement results along the equator. Here, the results do not need to be recorded because the corresponding paths only differ by a great circle and thus by a phase of 2π. However, the measurement results are important for measurements along the Z-axis, and the probability of completing the process from north → south → north is 1 / 4 (for ).

[0106] Obtaining an incorrect measurement result leads to a contribution which re-introduces an unwanted pre-factor that depends on τ z . This hinders the effective implementation of the forced measurement scheme because the terms that depend on τ z for incorrect measurement results need to be appropriately cancelled. However, this protocol produces the implementation of the geometric decoupling scheme with a probability of 2 -2N , which is not affected by unwanted couplings and records the measurement results along the Z-axis, and when this occurs, it is also known. In the next section, a discussion of how to use the hybrid evolution measurement scheme to thoroughly increase the probability of obtaining correct measurement results is provided.

[0107] Figure 4 is a schematic block diagram 400 that visualizes an example hybrid protocol. The lines represent the evolution of the state, and the arrows of the same color represent the corresponding Hamiltonian terms H 2j-1 and H 2j . For illustrative purposes, the reverse evolution is illustrated as moving π from the south pole to the north pole, corresponding to the application of -H 2j . The sign of H does not matter because it results in a phase difference of 2π. At the end of each evolution stage, a projection is performed towards the south pole or the north pole with a high success probability, as indicated by the red dots.

[0108] C. Hybrid Evolution and Measurement Scheme

[0109] By combining measurement with the dynamic evolution of the system, the problem of the low success probability of only measuring the north / south projection scheme can be avoided. During the process of this proposal, the projection measurements along the north and south poles are supplemented by a free (non-adiabatic) evolution that transfers the states of the four MZMs between the two poles. By combining measurement with free evolution, the probability of measurement error can be thoroughly reduced while eliminating the possibility of static machine errors (e.g., the case of θ = 0 from Equation (11)).

[0110] An example embodiment of this process is as follows. Just as in the only measurement process, the evolution can be decomposed into a series of wedge flows that flow from the north pole to the south pole and back respectively through azimuth angles and . The process of the j-th wedge first projects the qubit onto its north pole Pzτ Next, ideally, evolve the qubit with the Hamiltonian:

[0111]

[0112] where and σ = {σ x , σ y , σ z}. Note that, compared to the adiabatic evolution scheme, all couplings are turned off after projecting to the north pole at the start of the system. Thus, turning on the above Hamiltonian is non-adiabatic and behaves like a perpendicular magnetic field where the "rotation" of the qubit Bloch sphere precesses freely. This precession will evolve the state along a great arch intersecting the equator at until it reaches the south pole where should be measured as 1. To this end, the Hamiltonian (16) can be turned on for a time T = π(1 + 2n) / ω. This may seem like it has to be fine-tuned this time, but that is not the case. The next measurement will precisely align the qubit with the south pole, thus correcting for over- or under-rotation. As shown below, the main advantage of precisely tuning the free precession time is to increase the success probability of the process. After projecting to the south pole, the Hamiltonian

[0113]

[0114] can be applied for a time T = π(1 + 2n′) / ω. This will return the qubit near the north pole where a Z-based measurement will result in a possible projection to P zτ . Note that the negative sign in formula (17) is not necessary and it is chosen for convenience in aligning with the wedge in Figure 4 . Shifting ω → -ω results in an essentially equivalent path differing by a 2π phase.

[0115] Now, consider calculating the evolution of the qubit state in the above single-wedge process. According to the Pauli matrices, the operator is applied to the qubit readout according to the Pauli matrices:

[0116] Π hyb = P z τe -iTHj+1 P z -τe -iTHj P z τ (18)

[0117] At this point, it can be assumed that the Hamiltonian H j is not perfect, where a small part of iγ x γ y = σ z is mixed in due to unavoidable external couplings, as in formula (12):

[0118]

[0119] And similarly for any j.

[0120] One aspect of the method relies on projections onto the north and south poles before and after free evolution. Only the free evolution part of the flipped σ z survives the measurement. Using e -iTjHj = cos(ωT j / 2) - ih j ·σ sin(ωT j / 2) gives:

[0121]

[0122] Which is further reduced to:

[0123]

[0124] And finally

[0125]

[0126] Where

[0127]

[0128] Where the projection onto τ z σ z = -1, P zτ is used. Thus, it can be concluded that the state of the qubit enters only as a phase shift . Thus, it can be observed that the hybrid protocol allows the realization of the same projection as the measurement-only approach (see Equation (15)), but with the advantage of increasing the success probability. This makes it possible to implement the full Chebyshev protocol, as discussed in Section IVB3, with a higher probability, which supports a reasonable number of turning points (see the simulations in Section IVD).

[0129] As mentioned above, an exact phase rotation can be obtained without fine-tuning. T j and the error in the external coupling only suppress the probability p 2 j，j+1 of obtaining the correct measurement result (e.g., the distribution of the results of alternating iγ0γ z = ±1). If is small, as expected, and T j，j+1 can be adjusted to be close to their desired values, then this probability will be close to 1. Since one knows the measurement results, a success probability < 1 does not affect the fidelity of the implemented gate, it only increases the waiting time until a run with all the desired measurement results is achieved.

[0130] Note that since only the measurement and mixing scheme achieves similar (e.g., substantially the same) projections (15) and (22), the mixing scheme is also robust against unintended measurements of the (partial) projection system into the eigenstates of the Hamiltonian H j The latter effect only manifests itself in the change of the success probability of ultimately reaching formula (15) in the limit of strong measurement. Herein, such partial measurements in the eigenbasis of the Hamiltonian will be referred to as parallel dissipation, since they are caused by the system-environment coupling ∝H j induced. Note that measurements (or dissipation) acting along vectors perpendicular to H j will result in remaining decoherence. One advantage of the mixing scheme is that it is robust against the main sources of dissipation.

[0131] D. Numerical

[0132] In this section, the performance of the hybrid measurement scheme will be considered. The models of both, the free evolution described by the Hamiltonian similar to formula (16) and the effect of the measurement, are also considered. Measurements corresponding to the north and south poles can be implemented in a topologically protected manner. Thus, they are respectively described by the projections P zτ and The reduction of the trace norm of the system density matrix then quantifies the success probability of finding the measurement result corresponding to the projection.

[0133] For the evolution of qubits between the two poles, the environment can measure the state of the auxiliary qubits that cause decoherence. Similarly, the unprotected measurements along the equator, which are part of the implementation of only measurement, can also be modeled by decoherence, since no knowledge of their measurement results is required. This allows one to describe hybrid protocols with dissipation and only measurement protocols on the same basis, due to environmental noise along the measurement axis where the time evolution of the density matrix can be transformed into the form of the Lindblad master formula,

[0134]

[0135] where where Γ is the corresponding phase shift rate. The above master formula results from the system-environment coupling H SE = 1·σΦ / 2 after integrating out the environmental degrees of freedom assumed to be short-time correlated <Φ(t)Φ(0)> E = 2Γδ(t). Note that the assumption of a short-time correlated environment describes the worst-case scenario of the embodiments of the disclosed geometric decoupling scheme. The environmental noise on the time scale is longer than the repeated scans applied, and in fact, can be effectively eliminated by the Chebyshev protocol.

[0136] Figure 5 Figs. 500, 510, and 520 show several curves that illustrate simulation results demonstrating aspects of the disclosed technology. Figs. 500, 510, and 520 show simulation results for a single-wedge north / south sweep implementation of a π / 8 gate (turning points ). In general, timing affects the fidelity and success probability of a gate. Fidelity is defined as the overlap of the final state with the desired magic state in the case where a specific set of projections (here, north, south, north) is applied. The success probability quantifies the chance of obtaining the correct measurement result to achieve the above set of projections. Fig. 500 shows the case without dissipation (perfect case); Fig. 510 shows the case with dissipation 1 = h, Γ = 0.5ω / π, external coupling ; and Fig. 520 shows the case of misaligned dissipation (where the loss of fidelity is shown by the lower line). For comparison, the dashed line in Fig. 510 represents the success probability due solely to the phase shift (for simplicity, ) in the case of only the measurement process (1 - exp(-Γt)) / 2.

[0137] 1. Single-Wedge Example

[0138] The interaction between coherent and incoherent evolution can be demonstrated using a hybrid evolution corresponding to a single wedge tracing the Bloch sphere, see Figure 4 . In particular, consider the protocol that starts from the eigenstate of P zτ (north pole), projects onto P x (equator, ), and then projects onto (south pole). All measurements can be performed in a topologically protected manner, and forced measurements can be used to obtain the required measurement results. Thus, post-selection can be performed by re-normalizing the density matrix after applying the projection. The resulting state serves as the initial state evolving according to Equation (24). Then, by introducing a combination of coherent and incoherent couplings of Majorana modes within time T, the measurement process is modeled. Finally, at the end of the evolution, the system is projected back onto P zτ (north pole).

[0139] Let us first consider the fully coherent implementation of the π / 8 gate ( and Γ = 0 (see Figure 5 ), as described by Equation (22)). This implements the π / 8 gate with no doubt that ωT = (2n + 1)π. Departing from this perfect timing, the success probability of projecting the system onto the north pole decreases. However, if the projection onto the north pole is successful, the resulting state is a perfect magic state and no fidelity is lost.

[0140] Next, consider adding a finite external coupling and parallel dissipation 1 = h, Γ ≠ 0 (see 510). The external coupling limits the success probability to values < 1. Additionally, the oscillations will decay to a 50% success probability at a rate of Γ, resulting in a long time to reach the limit of just measurement.

[0141] Finally, when the coherent and incoherent parts of the evolution are not aligned h ≠ 1, the post-selected final state of the system becomes mixed. This irreversible decoherence causes the fidelity to increase over time. As expected, the highest fidelity can be achieved when ωT = π is approached.

[0142] E. Summary

[0143] The problem of implementing protected magic gates in Majorana systems remains a key challenge in the field. The gate is thought to require very precise control of Majorana couplings or an expensive distillation process. See J.D. Sau, S. Tewari, and S.D. Das Sarma, “Universal quantum computation in a semiconductor quantum wire network,” Phys. Rev. A 82, 052322 (2010), arXiv:1007.4204. In the present disclosure, it has been shown that a sequence of measurements and free evolution applied to four MZMs eliminates the need for fine-tuning and the adverse effects of all low-frequency noise. Thus, the only remaining source of error is high-frequency fluctuations, which cause the device to vary on a time scale shorter than the time required to complete one cycle.

[0144] The hybrid approach provides a significant simplification of the scheme proposed in T. Karzig, Y. Oreg, G. Refael, and M.H. Freedman, “Universal Geometric Path to a Robust Majorana Magic Gate,” Phys. Rev. X 6, 031019 (2016), arXiv:1511.05161. In fact, the latter includes echoes in the Majorana operations, which are designed to eliminate residual dynamical effects due to some inevitable couplings between MZMs. The echoes increase the vulnerability of the gate to noise that acts on a time scale faster than the duration of the entire decoupling scheme. The hybrid approach manages to absolutely eliminate the effects of unwanted couplings and thus removes the need for performing an echo phase. V. General exemplary embodiments of the disclosed technology

[0145] Figure 10FIG. 1000 is a flow chart showing a generalized exemplary embodiment for implementing an embodiment of the disclosed technology. The specific operations and the order of operations should not be construed as restrictive, as they may be performed individually or in any combination, sub-combination, and / or order with respect to each other. Additionally, the operations shown may be performed in conjunction with one or more other operations. In particular, FIG. 1000 illustrates a method for implementing a π / 8 phase gate in a quantum computing device.

[0146] At 1002, the quantum state of a quantum circuit configured to implement a π / 8 phase gate in a quantum computing device is changed from an initial state to a target state using a hybrid measurement scheme. In the illustrated embodiment, the hybrid measurement scheme includes: at 1010, applying one or more measurements to the quantum state that project the quantum state onto the target state; and at 1012, applying one or more adiabatic or non-adiabatic techniques that evolve the quantum state towards the target state.

[0147] In certain embodiments, the quantum computing device is a topologically protected quantum computing device. In some embodiments, the hybrid measurement scheme reduces timing noise, slow parameter noise, dynamic phase noise, and / or parallel dissipation. In certain embodiments, one or more measurements of the quantum state are applied between applications of any of the following: (a) two non-adiabatic techniques in a non-adiabatic technique; or (b) two adiabatic techniques in an adiabatic technique. In some embodiments, the application of one or more measurements of the quantum state includes general geometric decoupling of the quantum circuit. For example, general geometric decoupling may be performed by applying a continuous projection operator to the quantum circuit. In some examples, the continuous projection operator is applied at a turning point of a geometric decoupling trajectory, and the turning point is determined using Chebyshev polynomials. In further examples, general geometric decoupling of the quantum circuit maps to multiple parameter scans across the poles of the unit sphere.

[0148] Any of the embodiments disclosed above may be implemented as part of a system that includes: a quantum computing device that includes a quantum circuit; and a classical computing device that communicates with the quantum computing device and is adapted to perform any of the disclosed methods.

[0149] Any of the embodiments disclosed above may also be implemented by one or more computer-readable media storing computer-executable instructions that, when executed by a classical computer, cause the classical computer to perform a method of controlling a quantum computing device in accordance with any of the disclosed methods.

[0150] VI. Exemplary Computing Environments

[0151] Figure 6FIG. illustrates a generalized example of a suitable classical computing environment 600 in which several embodiments described therein can be implemented. Computing environment 600 is not intended to imply any limitation as to the scope of use or functionality of the disclosed technology, as the techniques and tools described herein can be implemented in a variety of general-purpose or special-purpose environments with computing hardware.

[0152] Referring Figure 6 , computing environment 600 includes at least one processing device 610 and a memory 620. In Figure 6 , this most basic configuration 630 is enclosed within the dashed line. The processing device 610 (e.g., a CPU or a microprocessor) executes computer-executable instructions. In a multiprocessing system, multiple processing devices execute computer-executable instructions to increase processing power. The memory 620 can be volatile memory (e.g., registers, caches, RAM, DRAM, SRAM), non-volatile memory (e.g., ROM, EEPROM, flash memory), or some combination of both. The memory 620 stores software 680 that implements tools for performing any of the quantum circuit control techniques disclosed herein. The memory 620 can also store software 680 for synthesizing, generating (or compiling), and / or controlling quantum circuits as described herein.

[0153] The computing environment can have additional features. For example, computing environment 600 includes a storage device 640, one or more input devices 650, one or more output devices 660, and one or more communication connections 670. An interconnection mechanism (not shown), such as a bus, a controller, or a network, interconnects the components of computing environment 600. Typically, an operating system software (not shown) provides an operating environment for other software executing in computing environment 600 and coordinates the activities of the components of computing environment 600.

[0154] The storage device 640 can be removable or non-removable and includes one or more disks (e.g., a hard disk drive), a solid-state drive (e.g., a flash drive), magnetic tape or cassette, a CD-ROM, a DVD, or any other tangible non-volatile storage medium that can be used to store information and can be accessed within computing environment 600. The storage device 640 can also store instructions for software 680 that implements any of the quantum circuit control mechanisms disclosed herein. The storage device 640 can also store instructions for software 680 for synthesizing, generating (or compiling), and / or controlling quantum circuits as described herein.

[0155] (Multiple) input devices 650 can be touch input devices such as a keyboard, a touch screen, a mouse, a pen, a trackball, a voice input device, a scanning device, or another device that provides input to the computing environment 600. (Multiple) output devices 660 can be a display device (e.g., a computer monitor, a laptop computer display, a smart phone display, a tablet computer display, a netbook display, or a touch screen), a printer, a speaker, or another device that provides output from the computing environment 600.

[0156] (Multiple) communication connections 670 support communication with another computing entity via a communication medium. The communication medium conveys information such as computer-executable instructions or other data in a modulated data signal. A modulated data signal is a signal whose one or more characteristics are set or changed in such a manner as to encode information in the signal. By way of example and not limitation, the communication medium includes wired or wireless technologies implemented using electrical, optical, radio frequency, infrared, acoustic, or other carriers.

[0157] As described above, various methods, quantum circuit control techniques, or compilation / synthesis techniques can be described in the general context of computer-readable instructions stored on one or more computer-readable media. A computer-readable medium is any available medium (e.g., a memory or a storage device) that can be accessed within or by a computing environment. The computer-readable medium includes tangible computer-readable memory or storage devices such as memory 620 and / or storage device 640, and does not include a propagated carrier wave or signal itself (tangible computer-readable memory or storage devices do not include a propagated carrier wave or signal itself).

[0158] Various embodiments of the methods disclosed herein can also be described in the general context of computer-executable instructions (such as those included in program modules) executed by a processor in a computing environment. Generally, program modules include routines, programs, libraries, objects, classes, components, data structures, etc. that perform particular tasks or implement particular abstract data types. In various embodiments, the functionality of program modules can be combined or split as needed among program modules. The computer-executable instructions for program modules can be executed in a local or distributed computing environment.

[0159] An example of a possible network topology 700 (e.g., a client-server network) for implementing a system according to the disclosed technology is depicted in Figure 7 A networked computing device 720 can be, for example, a computer running a browser or other software connected to a network 712. The computing device 720 can have as Figure 6The computer architectures shown and discussed above. Computing device 720 is not limited to a traditional personal computer, but may include other computing hardware configured to connect to and communicate with network 712 (e.g., smart phones, laptop computers, tablet computers, or other mobile computing devices, servers, network devices, specialized devices, etc.). Additionally, computing device 720 may include an FPGA or other programmable logic device. In the illustrated embodiment, computing device 720 is configured to communicate with computing device 730 (e.g., a remote server, such as a server in a cloud computing environment) via network 712. In the illustrated embodiment, computing device 720 is configured to transmit input data to computing device 730, and computing device 730 is configured to implement quantum circuit control techniques according to any of the disclosed embodiments and / or circuit generation or compilation / synthesis methods for generating quantum circuits for use with any of the techniques disclosed herein. Computing device 730 may output results to computing device 720. Any data received from computing device 730 may be stored or displayed on computing device 720 (e.g., displayed as data at a graphical user interface or web page at computing device 720). In the illustrated embodiment, the illustrated network 712 may be implemented as a local area network (LAN) using wired networking (e.g., Ethernet IEEE standard 802.3 or other suitable standard), or wireless networking (e.g., one of IEEE standards 802.11a, 802.11b, 802.11g, or 802.11n or other suitable standard). Alternatively, at least a portion of network 712 may be the Internet or a similar public network and operate using a suitable protocol (e.g., the HTTP protocol).

[0160] Another example of a possible network topology 800 (e.g., a distributed computing environment) for implementing a system according to the disclosed technology is in Figure 8 depicted. The networked computing device 820 may be, for example, a computer running a browser or other software connected to network 812. The computing device 820 may have as Figure 6The computer architectures shown and discussed above. In the illustrated embodiment, the computing device 820 is configured to communicate via the network 812 with a plurality of computing devices 830, 831, 832 (e.g., remote servers or other distributed computing devices, such as one or more servers in a cloud computing environment). In the illustrated embodiment, each of the computing devices 830, 831, 832 in the computing environment 800 is used to perform at least a portion of a quantum circuit control technique according to any of the disclosed embodiments and / or a circuit generation or compilation / synthesis method for generating a quantum circuit for use with any of the techniques disclosed herein. In other words, the computing devices 830, 831, 832 form a distributed computing environment in which the quantum circuit control and / or generation / compilation / synthesis processes are shared across multiple computing devices. The computing device 820 is configured to transmit input data to the computing devices 830, 831, 832, and the computing devices 830, 831, 832 are configured to distributively implement processes such as the execution of any of the disclosed methods or the creation of any of the disclosed circuits and are configured to provide results to the computing device 820. Any data received from the computing devices 830, 831, 832 may be stored or displayed on the computing device 820 (e.g., displayed as data at a graphical user interface or web page at the computing device 820). The illustrated network 812 may be any of the networks discussed above with respect to Figure 7 and discussed.

[0161] Referring Figure 9 , an exemplary system for implementing the disclosed techniques includes a computing environment 900. In the computing environment 1100, a compiled quantum computer circuit description (including a quantum circuit generated and / or supported by any of the quantum circuit control techniques disclosed herein) may be used to program (or configure) one or more quantum processing units such that the (multiple) quantum processing units implement the circuit described by the quantum computer circuit description.

[0162] The environment 900 includes one or more quantum processing units 902 and one or more readout devices 908. The quantum processing unit executes a quantum circuit pre-compiled and described by a quantum computer circuit description. The (multiple) quantum processing units can be, but are not limited to, one or more of the following: (a) a superconducting quantum computer; (b) an ion trap quantum computer; (c) a fault-tolerant architecture for quantum computing; and / or (d) a topological quantum architecture (e.g., a topological quantum computing device using Majorana zero modes). The pre-compiled quantum circuit including any of the circuits disclosed herein can be sent to (or otherwise applied to) the (multiple) quantum processing units via control lines 906 under the control of a quantum processor controller 920. The quantum processor controller (QP controller) 920 can operate with a classical processor 910 (e.g., having an architecture as described above with respect to Figure 6 to implement the desired quantum computing process. In the example shown, the QP controller 920 also implements the desired quantum computing process via one or more QP sub-controllers 904, which are specifically adapted to control the corresponding quantum processors in the (multiple) quantum processors 902. For example, in one example, the quantum controller 920 facilitates the implementation of the compiled quantum circuit by sending instructions to one or more memories (e.g., cryogenic memories), which then pass the instructions to the (multiple) cryogenic control units (e.g., the (multiple) QP sub-controllers 904), and the QP sub-controllers 904 transmit a pulse sequence representing gates, for example, to the (multiple) quantum processing units 902 for implementation. In other examples, the (multiple) QP controllers 920 and the (multiple) QP sub-controllers 904 operate to provide appropriate magnetic fields, encoding operations, or other such control signals to the (multiple) quantum processors to implement the operations described by the compiled quantum computer circuit description. The (multiple) quantum controllers can further interact with the readout device 908 to assist in controlling and implementing the desired quantum computing process (e.g., by reading or measuring data results from the quantum processing unit once available, etc.).

[0163] Reference Figure 9 , compilation is the process of translating a high-level description of a quantum algorithm into a quantum computer circuit description that includes a sequence of quantum operations or gates, which can include circuits as disclosed herein. Compilation can be performed by a compiler 922 using the classical processor 910 of the environment 900 (e.g., as shown in Figure 6 ), and the processor 910 loads the high-level description from a memory or storage device 912 and stores the resulting quantum computer circuit description in the memory or storage device 912.

[0164] In other embodiments, compilation and / or verification can be performed by a remote computer 960 (e.g., having an architecture as described above with respect to Figure 6executed remotely by a computer of the described computing environment, which stores the resulting quantum computer circuit description in one or more memories or storage devices 962 and transmits the quantum computer circuit description to the computing environment 900 for implementation in the (multiple) quantum processing units 902. Further, the remote computer 900 may store the high-level description in a memory or storage device 962 and transmit the high-level description to the computing environment 900 for compilation and use with the (multiple) quantum processors. In any of these scenarios, the results from the computation executed by the (multiple) quantum processors may be transmitted to the remote computer after and / or during the computation process. Further still, the remote computer may communicate with the (multiple) QP controllers 920 such that the quantum computing process (including any compilation, verification, and QP control processes) may be remotely controlled by the remote computer 960. Generally, the remote computer 960 communicates with the (multiple) QP controllers 920, compiler / synthesizer 922, and / or verification tool 923 via a communication connection 950.

[0165] In a particular embodiment, the environment 900 may be a cloud computing environment that provides the quantum processing resources of the environment 900 to one or more remote computers (such as the remote computer 960) via a suitable network (which may include the Internet).

[0166] VII. Conclusion

[0167] The principles of the disclosed technology have been described and illustrated with reference to the shown embodiments, and it will be recognized that the shown embodiments may be modified in arrangement and detail without departing from such principles. For example, elements of the shown embodiments that are shown in software may be implemented in hardware and vice versa. Additionally, techniques from any example may be combined with techniques described in any one or more of the other examples. It will be appreciated that processes and functions such as those described with reference to the shown examples may be implemented in a single hardware or software module or separate modules may be provided. The above particular arrangements are provided for convenience of illustration and other arrangements may be used.

Claims

1. A method for implementing a π / 8 phase gate in a quantum computing device, comprising: Changing the quantum state of a quantum circuit configured to implement the π / 8 phase gate in the quantum computing device from an initial state to a target state using a hybrid measurement scheme that reduces timing noise, slow parameter noise, dynamic phase noise, and / or parallel dissipation, the hybrid measurement scheme comprising: Applying a plurality of measurements to the quantum state, the plurality of measurements projecting the quantum state towards the target state, wherein the application of the plurality of measurements to the quantum state includes a general geometric decoupling of the quantum circuit, the general geometric decoupling including applying a continuous projection operator to the quantum circuit, the continuous projection operator being applied at turning points of a geometric decoupling trajectory, and wherein the turning points are determined using Chebyshev polynomials; and Applying non-adiabatic Hamiltonian evolution that evolves the quantum state towards the target state, wherein the non-adiabatic Hamiltonian evolution is applied between the plurality of measurements.

2. The method according to claim 1, wherein the quantum computing device is a topologically protected quantum computing device.

3. The method according to claim 1, wherein the non-adiabatic Hamiltonian evolution comprises a plurality of non-adiabatic Hamiltonian evolutions, and the plurality of measurements of the quantum state are applied between the applications of two non-adiabatic Hamiltonian evolutions of the plurality of non-adiabatic Hamiltonian evolutions.

4. The method according to claim 1, wherein the general geometric decoupling of the quantum circuit maps to a plurality of parameter scans across the poles of the unit sphere.

5. A system for quantum computing, comprising: A quantum computing device including a quantum circuit; And A classical computing device that communicates with the quantum computing device and is adapted to execute a method that includes: Changing the quantum state of a quantum circuit configured to implement a π / 8 phase gate in the quantum computing device from an initial state to a target state using a hybrid measurement scheme that reduces timing noise, slow parameter noise, dynamic phase noise, and / or parallel dissipation, the hybrid measurement scheme comprising: Applying a plurality of measurements to the quantum state, the plurality of measurements projecting the quantum state towards the target state, wherein the application of the plurality of measurements to the quantum state includes a general geometric decoupling of the quantum circuit, the general geometric decoupling including applying a continuous projection operator to the quantum circuit, the continuous projection operator being applied at turning points of a geometric decoupling trajectory, and wherein the turning points are determined using Chebyshev polynomials; and Applying non-adiabatic Hamiltonian evolution that evolves the quantum state towards the target state, wherein the non-adiabatic Hamiltonian evolution is applied between the plurality of measurements.

6. The system according to claim 5, wherein the quantum computing device is a topologically protected quantum computing device.

7. The system according to claim 5, wherein the non-adiabatic Hamiltonian evolution comprises a plurality of non-adiabatic Hamiltonian evolutions, and the plurality of measurements of the quantum state are applied between the applications of two non-adiabatic Hamiltonian evolutions of the plurality of non-adiabatic Hamiltonian evolutions.

8. The system according to claim 5, wherein the general geometric decoupling of the quantum circuit maps to a plurality of parameter scans across the poles of the unit sphere.

9. A computer-readable medium storing computer-executable instructions that, when executed by a classical computer, cause the classical computer to execute a method for controlling a quantum computing device, the method comprising: changing a quantum state of a quantum circuit configured to implement a π / 8 phase gate in the quantum computing device from an initial state to a target state using a hybrid measurement scheme that reduces timing noise, slow parameter noise, dynamic phase noise, and / or parallel dissipation, the hybrid measurement scheme comprising: applying a plurality of measurements to the quantum state, the plurality of measurements projecting the quantum state towards the target state, wherein the application of the plurality of measurements of the quantum state comprises general geometric decoupling of the quantum circuit, the general geometric decoupling comprising applying a continuous projection operator to the quantum circuit, the continuous projection operator being applied at turning points of a geometric decoupling trajectory, and wherein the turning points are determined using Chebyshev polynomials; and applying non-adiabatic Hamiltonian evolution, the Hamiltonian evolution evolving the quantum state towards the target state, wherein the non-adiabatic Hamiltonian evolution is applied between the plurality of measurements.

10. The computer-readable medium according to claim 9, wherein the quantum computing device is a topological quantum computing device.

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