Measurement-only Majorana-based surface code architecture

Abstract

The quantum device includes a syndrome measurement circuit that implements a correction code using a plurality of Majorana qubit islands. The syndrome measurement circuit is configured to perform a syndrome measurement by performing a series of measurement-only operations, each one involving at most two Majorana qubit islands.

Description

[Background technology]

[0001] background

[0001] A Majorana fermion is a fermion that is its own antiparticle. The zero modes of Majorana fermions can be localized at defects such as domain boundaries or vertices. Such Majorana zero modes (MZMs) are interesting for quantum computing because they can host a non-localized topologically protected state space. Several recent experiments have demonstrated that MZMs can exist at both ends of a one-dimensional semiconductor wire attached to a superconductor. MZMs are currently being investigated as potential building blocks for qubits, useful for performing topological quantum computing.

[0002]

[0002] One approach to topological quantum computing using MZMs utilizes a scheme in which MZMs move around each other, the movements producing a set of transformations on non-localized state spaces (called "braid exchanges" or "braid transformations") that may represent corresponding computational gates acting on qubit states encoded in these state spaces. For example, in a particular encoding scheme, moving one quasiparticle around another clockwise can correspond to a NOT gate (Pauli X) on the qubits. To read out the qubits, the quasiparticle pair can be forced to collide together on a nanowire so that the result can be measured. It has been found that braid transformations of MZMs can be used to generate a set of Clifford gates acting on a system of qubits.

[0003]

[0003] Another approach to topological quantum computing using MZMs utilizes an anionic analogue of quantum state teleportation to produce braid transformations without physically transporting the MZMs around each other. This particular approach represents an example of "measurement-only topological quantum computing". In this approach, quantum state teleportation is realized by projective measurement of the connected fermion parity of the MZM pair together with the use of an auxiliary MZM. Repeated application of quantum state teleportation can have the same effect as the braid swap of two MZMs in the above approach where the MZMs are physically moved around. Hence, a sequence of MZM measurements is performed to implement a set of Clifford quantum gates. The technology described herein relates to this latter approach. Summary of the Invention [Means for solving the problem]

[0004] overview According to one embodiment, a quantum device includes a syndrome measurement circuit that implements an error-correcting code using a plurality of Majorana qubit islands, the syndrome measurement circuit being configured to perform a syndrome measurement by performing a series of measurement-only operations, each one of which involves at most two of the Majorana quantum islands. [Brief description of the drawings]

[0005] BRIEF DESCRIPTION OF THE DRAWINGS [Figure 1]

[0005] An example quantum computing system is presented that performs error correction via a quantum measurement circuit that implements a surface code using a measurement-only Majorana-based architecture. [Figure 2A]

[0006] 1 shows an example of a two-sided Majorana hexon architecture that can be used to perform efficient surface code syndrome measurements. [Figure 2B]

[0007] 1 shows an example of a one-sided Majorana hexon architecture that can be used to perform efficient surface code syndrome measurements. [Diagram 3]

[0008] An example of a surface code architecture formed by mesoscopic superconducting islands including Majorana hexons and Majorana tetrons is shown. [Figure 4]

[0009] 1 illustrates further aspects of an example of a surface code architecture that includes Majorana hexons and Majorana tetron islands. [Diagram 5]

[0010] FIG. 5 and the following discussion are intended to provide a brief general description of an exemplary computing environment in which the disclosed technology may be implemented. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0006] Detailed Description

[0011] Quantum computers may perform error correction using quantum error correction codes (QECCs). QECCs encode logical qubits into a collection of physical qubits such that the error rate of the logical qubits is lower than the physical error rate. Several error correction protocols have been proposed in recent years. QECCs enable fault-tolerant quantum computing, at the expense of an increased number of physical qubits, as long as the physical error rate is below an acceptable threshold.

[0007]

[0012] During quantum processing, entropy from data qubits that encode the data to be protected is converted into auxiliary qubits that can be discarded and are positioned to interact with the data qubits such that errors can be detected by measuring the auxiliary bits and corrected using a decoding unit.

[0008]

[0013] An important class of error-correcting codes is stabilizer codes. For stabilizer codes, a logical qubit is defined as the simultaneous +1 eigenspace of some number of commuting multi-qubit Pauli operators, called stabilizers. By repeatedly measuring the quantum system using a full set of commuting stabilizers, the data bits associated with each ancillary qubit are forced to be in simultaneous and unique eigenstates of all stabilizers, allowing the stabilizers to be measured without perturbing the system. If the measurement result changes, this corresponds to one or more qubit errors, and the quantum state is projected by the measurement into a different stabilizer eigenstate. Errors are detected by repeatedly measuring the stabilizers, and deviations from the expected result of +1 indicate an error.

[0009]

[0014] Within the class of stabilizer codes, surface codes are considered the most promising QECCs for fault-tolerant quantum computation. The simplest realization is defined on a rectangular lattice of qubits, with the plaquette divided into two sublattices in a checkerboard pattern. There is one stabilizer for each plaquette: in one sublattice, it is given by the product of four Pauli X operators on the data qubits around the plaquette; in the other sublattice, it is given by the corresponding product of four Pauli Z operators.

[0010]

[0015] In the most practical proposal to implement surface codes, a stabiliser measurement (e.g. the product of four Pauli operators acting on data qubits around a plaquette) is achieved by adding an additional ancillary qubit to each plaquette, entangling it in a particular way with the neighbouring data qubits, and finally performing a single-qubit measurement on the ancillary qubit.

[0011]

[0016] Since measurements of Pauli operators become topologically protected parity measurements in MZM-based architectures, Pauli stabilizer codes map ideally to such architectures. According to one embodiment of the disclosed technology, surface codes are implemented in quantum devices via measurement circuits formed of mesoscopic superconducting islands, more specifically referred to herein as Majorana hexons (or simply "hexons") and Majorana tetrons (or simply "tetrons"). In implementing surface codes to correct measurement errors, optimized stabilizer measurements are made by measuring sequences of topologically protected Majorana-based qubits located in nearest-neighbor mesoscopic superconducting islands. According to the proposed approach, this stabilizer measurement can be performed using only measurements involving two or four MZMs at a time (2MZM measurements or 4MZM measurements), with each such measurement involving MZMs in a single island or two nearest-neighbor islands. Due to the reduced number of measurements and the proximity of the MZMs involved in the measurements, this stabilizer measurement can be performed with much less processing resources than traditional measurement-only approaches.

[0012]

[0017] Below, in Figure 1 and the associated description, we provide an overview of the types of quantum computing systems and measurement circuits that may be implemented using the proposed measurement-only Majorana-based surface code architecture. In Figure 2 and the associated description, we discuss the individual building blocks of this architecture, while the discussion of Figures 3 and 4 introduces details about the proposed surface code architecture and an exemplary optimized use of such an architecture.

[0013]

[0018] FIG. 1 illustrates an example quantum computing system 100 that performs error correction via a syndrome measurement circuit 114 that implements a surface code using a measurement-only Majorana-based architecture.

[0014]

[0019] Quantum computing system 100 includes controller 102, which performs computations by manipulating qubits in quantum register 108. To enable fault-tolerant quantum computing in quantum computing system 100, readout device 112 includes syndrome measurement circuit 114, which applies surface codes to qubits in qubit register 108. Syndrome measurement circuit 114 uses additional qubits, known as "ancillary qubits," to perform computations involving data qubits in quantum register 108. Syndrome measurement circuit 114 performs measurements of ancillary qubits in the quantum computer to extract syndromes that provide measured information about errors (faults). To avoid accumulation of errors during quantum computing, syndrome data is constantly measured, generating r syndrome bits for each syndrome measurement round. In one implementation, syndrome data is measured as frequently as once every 1 μs.

[0015]

[0020] The syndrome data output by the syndrome measurement circuit 114 is sent to a decoding unit 116, which implements one or more QECCs to analyze the syndrome data, detect the location of each error, and correct each error in the syndrome measurement circuit 114. The decoding unit 116 returns information to the controller 102 for correcting each detected error.

[0016]

[0021] View 142 shows, by way of example and not limitation, a portion of a surface code architecture used to form syndrome measurement circuit 114. The surface code architecture encodes combinations of data qubits (e.g., data qubit 118), represented by hollow circles, and ancillary qubits (e.g., ancillary qubit 120), represented by solid circles, arranged in a lattice-type structure defined by several plaquette (e.g., plaquette 122, 124), each of which includes a central ancillary qubit and four data qubits located at the corners.

[0017]

[0022] During quantum processing, entropy from data qubits that encode the data to be protected is converted to ancillary qubits (e.g., from the corners of the plaquette to the central ancillary qubit) that can be discarded after a measurement. This data conversion between the data qubits at the corners of each plaquette and the central ancillary qubit (e.g., ancillary qubit 120) is accomplished via a process that involves entangling the four corner data qubits and the central ancillary qubit and then performing a protection measurement on the central ancillary qubit. This process, referred to herein as a "stabilizer measurement," produces individual syndromes that are then transmitted to the decoding unit 116.

[0018]

[0023] The surface code performs two types of stabilizer measurements - X-type stabilizer measurements and Z-type stabilizer measurements. Each plaquette in the syndrome measurement circuit 114 is used to perform either X-type stabilizer measurements or Z-type stabilizer measurements. For example, a plaquette with a shaded surface (e.g., plaquette 124) may be used for Z-type stabilizer measurements, and an unshaded plaquette may be used for X-type stabilizer measurements. Under this architecture, the ancillary qubit at the center of each shaded plaquette is called the "Z measurement" qubit, while the ancillary qubit at the center of each unshaded plaquette is called the "X measurement" qubit. Each X and Z measurement qubit is coupled to four data qubits (e.g., four nearest hollow circles), and each data qubit is individually coupled to two Z and two X measurement qubits (e.g., four nearest solid circles).

[0019]

[0024] A stabilizer measurement of an individual plaquette 122 of the surface code is accompanied by a sequence of measurements that at least involves (1) performing a first measurement to initialize the central ancillary quantum bit in the computational state |0>, (2) performing four CNOT operations to entangle all five quantum bits on the plaquette 122, and (3) performing a projected measurement of the central ancillary quantum bit.

[0020]

[0025] In the plaquette used to perform the Z-type stabilizer measurements, the 4-CNOT measurement step (step (2) above) targets the measurement qubit using the four nearest-neighbor data qubits as controls, and the protection measurement targets the eigenstates

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[0021]

[0026] In different quantum platforms, the four CNOT operations of the surface code stabilizer measurement may be implemented in a variety of ways. In the architecture proposed herein, MZMs are used to implement the surface code. The ancillary qubits at the center of each plaquette (e.g., plaquette 122) are implemented by MZM structures known as Majorana hexons, referred to herein as "hexons," which include six MZMs. The data qubits at the corners of each plaquette are implemented by structures known as Majorana tetrons, referred to herein as "tetrons," which each include four MZMs. For example, view 130 in FIG. 1 shows Majorana island group 148, which includes central hexon 134 surrounded by four tetrons 136, 138, 140, and 144. Hexon 134 and tetrons 136, 138, 140, and 144 together form five data bits on plaquette 122. The stabilizer measurements of the surface code can be implemented by performing a series of connected fermion parity measurements involving a smaller number of MZMs in the Majorana island group 148. For example, the four CNOT operations of the stabilizer measurements can be performed using a series of 2-MZM and 4-MZM parity measurements. This measurement-only approach using hexons and tetrons allows for more efficient computations than previous approaches.

[0022]

[0027] 2A and 2B show example MZM hexon architectures 200, 202 that can be used to perform efficient surface code syndrome measurements. An MZM hexon (referred to herein simply as a "hexon") is a superconducting island that includes six MZMs (e.g., MZM 204), some of which are used to encode qubit states while others act as auxiliary degrees of freedom to facilitate measurement-based computations. The MZM hexon architecture 200 of FIG. 2A is a two-sided hexon, while the MZM hexon architecture 202 of FIG. 2B is a one-sided hexon.

[0023]

[0028] A qubit can also be formed from four MZMs (called a tetron when on an isolated superconducting island), but due to the absence of auxiliary MZMs, such a qubit is not capable of any topologically protected unitary gate operations by itself. On the other hand, in a hexon, the full set of single-qubit Clifford gates can be implemented with topological protection.

[0024]

[0029] According to one embodiment, the MZMs (e.g., MZMs 204, 206) contained in each hexon are connected by a topological superconductor wire (e.g., topological superconductor 208) and a spine made of trivial (s-wave) superconductor superconductors (e.g., superconductors 210, 214).

[0025]

[0030] In the double-sided architecture of FIG. 2A, three superconductor wires are connected by a central spine, and an MZM resides at both ends of each superconductor 208. In the single-sided architecture of FIG. 2B, six nanowires are connected by a spine 210 at one of the ends, and therefore an MZM resides only at the other end. An important advantage of these architectures is that the single qubit island is galvanically isolated (except for a weak coupling to the dot, as shown below), and therefore Coulomb interactions are limited by the island's finite charging energy E CThis helps to avoid quasiparticle poisoning (from the outside), because the probability of an electron tunneling in or out of the island from the outside is reduced by the charging energy E C and temperature ratio exp(-E C / K B T). The decoherence of the topologically protected state due to thermally excited quasiparticles on the island is exponentially suppressed at exp(-Δ / K B T), where Δ is the topological gap. The degeneracy splitting due to virtual tunneling of fermions between the MZMs is constrained by exp(-L / ξ), where L is the separation of the MZMs and ξ is the superconducting coherence length.

[0026]

[0031] According to the on-embodiment, projective measurements of the joint fermion parity of any two MZMs (2-MZM measurements) can be performed by enabling weakly coherent single-electron tunneling between the MZMs and adjacent quantum dots (e.g., quantum dots 214, 216), forming an interference loop. Projective measurements of the collective fermion parity of 2N-MZMs can be performed as well, but care must be taken to ensure that the interference loop includes all of the 2N MZMs, e.g., that fermions cannot pass directly between the various quantum dots involved. These couplings give rise to shifts in the energy spectrum and charge occupancy of the dots that depend on the fermion parity of the MZMs. These shifts can then be measured using established techniques developed for charge and spin qubits, such as charge sensing or quantum capacitance measurements. Importantly, the measurements are topologically protected, in the sense that the operators being measured are known to a correction that is exponentially small in the distance that separates the MZMs through the superconducting regions (nanowires and spines).

[0027]

[0032] Multiple hexons can be arranged in an array, and multi-qubit operations (e.g., CNOT operations used in surface code stabilizer measurements) can be performed by weakly coupling MZMs from different islands (e.g., different hexons or tetrons) to a common quantum dot. Because the coupling between the MZMs and the quantum dot is weak, charging energy protection against quasiparticle poisoning remains valid during such operations. This restricts the operators that can be measured to those that commute with the charging energy (or total parity) on each island, which is precisely measurements involving an even number of Majorana operators on each island.

[0028]

[0033] 3 shows an example surface code architecture 300 formed by mesoscopic superconducting islands of two different morphologies—Tetrons and Hexons—all arranged in a rectangular array. The hexons (e.g., hexons 302, 304) may have the same or similar individual structures as described above with respect to FIGS. 2A and 2B above, while the Tetrons (e.g., Tetrons 306, 308, 310, 312, 314, and 316) may be understood as structures including four MZMs each located at the end of a topological superconductor and attached to the same superconductor.

[0029]

[0034] Each Majorana hexon hosts six Majorana zero modes, which is the minimum number of MZMs that encode combinations of two qubits. Using one of the qubits as a data qubit and the others as auxiliary qubits facilitates arbitrary single-qubit Clifford operations on the data qubits in a measurement-only topologically protected manner.

[0030]

[0035] In contrast, each Majorana tetron hosts four Majorana zero modes, represented by the circles in the corners, which can be used to collectively encode a single computational qubit in the nonlocalized (topological) state space of the MZM - the joint fermionic parity of the MZM pair. Separately, a single Majorana tetron can be used to perform a Pauli operator measurement of a qubit, but not to enact a Clifford gate. However, it has been found that all n-qubit Clifford operations with topological protection can be performed using the joint parity measurements of four MZMs, two from one Majorana tetron and two from another.

[0031]

[0036] In the surface code architecture 300, the tetrons serve as data qubits, while the hexons are used as ancillary qubits that facilitate performing unitary operations and the X and Z stabilizer measurements. To avoid confusion between the term "ancillary qubit" used in reference to the second qubit encoded in the hexon and the term "ancillary qubit" used in reference to the qubit used to facilitate the stabilizer measurements in the surface code, the ancillary qubit of the surface code will be referred to as an "ancillary hexon" in the following description.

[0032]

[0037] In FIG. 3, the auxiliary hexons depicted on the light gray shaded plaquette correspond to auxiliary qubits in the surface code that facilitate the measurement of the X stabilizer. These hexons are referred to herein as “M X Similarly, the auxiliary hexons depicted on the dark grey shaded plaquette correspond to auxiliary qubits in the surface code that facilitate the measurement of the Z stabilizer. These hexons are referred to herein as "M Z It's called "Hexon."

[0033]

[0038] As discussed with respect to FIG. 1, each X-stabilizer and Z-stabilizer is achieved via initialization of a central ancillary qubit in its computational state |0>, followed by four CNOT operations, and a syndrome measurement circuit accompanied by a projective measurement.

[0034]

[0039] Due to the above properties of Majorana hexons and Majorana tetrons, the implementation of each CNOT operation can be performed using three Majorana tetrons, two Majorana hexons, or - in the proposed approach - one Majorana hexon and one Majorana tetron. This mixed approach is advantageous because it offers an optimized trade-off between efficiency and qubit quality. This is partly because Majorana tetrons exhibit a longer lifetime (better quality), but more Majorana tetrons are needed to perform the same quantum operation that can be performed by a single Majorana hexon. The proposed mixed hexon / tetron architecture not only lowers the total number of measurement operations required to perform a surface code syndrome measurement compared to existing approaches, but also provides a higher overall qubit quality.

[0035]

[0040] According to one embodiment, the surface code syndrome measurement circuitry is implemented for each ancillary hexon performing a series of connected fermion parity measurements on nearest-neighbor Majorana qubit islands, where each individual measurement in the series involves no more than two nearest-neighbor islands.

[0036]

[0041] Before introducing a specific exemplary measurement sequence that optimizes the efficiency of the surface code syndrome measurement circuitry, a discussion of hexon space states, operators, and notation is first presented below.

[0037] Single hexon state spaces and operators.

[0042] The MZMs in each hexon can be numerically designated 1, 2, 3, 4, 5, and 6. Similarly, the MZMs in each tetron can be numerically designated 1, 2, 3, 4. In the notation used in the following discussion, each of these MZMs is expressed as the Majorana fermion operator γ j These operators are expressed by the usual fermionic noncommutative relations {γ j ,γ k}=2δ jk For any ordered pair of j and k in the MZM, their connected fermion parity operator is iγ j Gamma k =-iγ k Gamma i which are the eigenvalues ​​p jk = ±1. (The convention in this paper differs slightly from that in reference

[15] .) The parity s = p jk The corresponding projection operator onto the subspace with = ±1 is

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[0038]

[0043] In this way, the ground state p 12 , p 34 , p 56can be used to represent the six MZM system in terms of fermion parity with some choice of how to pair together. Due to the finite charging energy of the islands, the system will generally only have ground states in either the even or odd collective fermion parity sectors, which can be tuned using gate voltages; without loss of generality, the system can be tuned to have ground states with even collective fermion parity, i.e., p 12 p 34 p 56 = +1, whereas states with odd collective parity can be assumed to be excited states associated with quasiparticle poisoning. Thus, the low-energy state space of the hexon is four-dimensional, and the ground state is │+,+,+〉 (3) │-,+,-〉=iγ2γ5│+,+,+〉, (4) │+,-,-〉=iγ4γ5│+,+,+〉 (5) │-,-,+〉=iγ2γ3│+,+,+〉 (6) It is.

[0039]

[0044] The first qubit is p 34 and the second qubit is p 12 If we view this as a two-qubit system encoded in the order 0,0,0,1,1,0,1,1, then the basis states are 0,0,0,1,1,0,1,1, in that order. We can then express the MZM parity operator in terms of the Pauli operators for these two qubits:

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[0040]

[0045] FIG. 3 introduces the numbering convention for hexon 316, in which MZM3 and 4 have definite linkage parity, e.g., p 34 = +1, and the computation qubit is p 12The remaining parity is encoded as p 56 =P 12 P 34 and so the auxiliary pair is p 34 = +1, the calculated ground state is |0>=|p 12 =p 56 =+>, |1>=|p 12 =p 56 =-> (9) and p 34 =-1, the calculated ground state is |0>=|p 12 =-p 56 =+>, |1>=|p 12 =-p 56 =-> (10) It is.

[0041]

[0046] Another way to look at this is that a hexon is a Majorana stabilizer code that encodes a single logical qubit into six MZMs. In this terminology, a logical qubit is defined as lying in a simultaneous +1 eigenspace of a group of operators called the stabilizer group. The logic gates that operate in this space are operators that commute with the stabilizer group but are not themselves stabilizers. In the case of a hexon, the stabilizer group is the total parity of the island, i 3 The logical Pauli operator is

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[0042] Auxiliary hexon syndrome measurement circuit

[0047] According to one embodiment, a single M X The syndrome measurement circuitry of Hexon 302 (e.g., measuring its logical qubits) involves the following sequence of operations: 1.M X Let us initialize the hexon 302 to the X = +1 state, and let the ancillary qubits of the hexon be in arbitrary but deterministic states (e.g., iγ1γ6 = +, iγ3γ4 = p34 in the state), 2. Apply a series of CNOT operations:

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[0043]

[0048] The effect of this series of steps is to

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[0044]

[0049] According to one embodiment, a single M Z The syndrome measurement circuitry of Hexon 304 involves the following sequence of operations: 1.M X We initialize the hexon qubit to the 0 (Z = +1) state, and the hexon ancillary qubits are in arbitrary but deterministic states (i.e., iγ1γ2 = +, iγ3γ4 = p 34 in the state), 2. Apply a set of CNOTs:

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[0045]

[0050] The most efficient computation of the above depends on an optimized compilation of these circuits (e.g. steps 1-3 in each of the above scenarios). There is no room for optimization since steps 1 and 3 above are simply measurements (two are needed for step 1 and one for step 3). Therefore, we can focus instead on step 2,

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[0046]

[0051] FIG. 4 shows LTC3111 that can be used to implement four CNOTs in each of the X-type and Z-type stabilizer measurement circuits. X Measuring circuit 410 and L Z Further details of the exemplary surface code architecture 400 are shown with respect to an exemplary representation of the measurement circuit 412 (see Equations 11 and 12 above). X and L Z The notation required to interpret the circuit is detailed below with respect to an exploded view 402 of the surface code architecture 400 and an exemplary MZM numbering scheme.

[0047]

[0052] The exploded view 402 shows six tetrons (labeled with the letters A, B, C, D, E, F, and G) and two hexons (M X and (M Z ) is shown. In this structure, M X The hexon is surrounded by tetrons A, B, C, and D, each of which acts as an individual data qubit. ZThe hexons are similarly surrounded by four tetrons E, F, A, and B, which act as data qubits. Within each hexon and tetron, the MZMs are numbered according to an exemplary numbering scheme. Majorana tetrons support modes 1, 2, 3, and 4, respectively, while Majorana hexons support modes 1, 2, 3, 4, 5, and 6.

[0048]

[0053] Each of the measurement circuits 410 and 412 in FIG. 4 is optimized to measure a series of four CNOTs (e.g., for measuring X- and Z-type syndrome measurement circuits) as described in Equations 11 and 12 above. X and L Z Specifically, the measurement circuit 410 provides a series of projective measurements of 2-MZM and 4-MZM joint fermion parity that can be performed to implement the following: X hexon), while the measurement circuit 412 measures the ancillary qubit (M Z A series of projection measurements for measuring the hexon are shown.

[0049]

[0054] The particular fermion parity measurement sequences illustrated by the measurement and measurement circuits 410 and 412 are intended to be illustrative, but not exclusive. In particular, the individual measurements have variable sequence lengths (number of measurements) and variable difficulty costs. X and L Z There are countless solutions to implement this. Measurement circuits 410 and 412, also shown in Table 1.0 below, represent one optimized approach derived from the following considerations.

[0050] [Table 1]

[0051] L X and L Z Measurement circuit description

[0055] Measurement circuit 410(L X) (also shown in Table 1.0 above), the first column is M X The second column corresponds to hexon, and the second column is M Z The first column corresponds to a hexon, and the next four columns correspond, from left to right, to the tetrons (A, B, C, D, E, and F) positioned as shown in exploded view 402. From top to bottom, eight measurement steps are shown in different rows. These steps are X and thus an auxiliary hexon (A) (representing an auxiliary qubit in each surface code plaquette) with four adjacent (B, C, D, and E) (representing the data qubits in each surface code plaquette).

[0052]

[0056] By way of example and not limitation, L X The first measurement step (first line of the measurement circuit 410) in X The second measurement step (line 2) is a 4-MZM measurement of the MZM2 and 4-connected fermion parities of the hexon. X This is accompanied by a 2-MZM measurement of the joint fermion parity of the MZM1 and 2 of the hexon. The third measurement step (line 3) involves the X This is accompanied by a 4-MZM measurement of the joint fermion parity of MZMs 1 and 3 of the hexon and MZMs 1 and 4 of the lower tetron (D). The fourth, fifth, sixth, seventh, and eighth measurement steps are further provided by measurement circuitry 410, consistent with the notation above.

[0053]

[0057] Measurement circuit 412(L Z ), the notation is for the measurement circuit 410 (L X ) is the same as described above. X and L ZEach of the measurement circuits (e.g., the four CNOT operations of the stabilizer) is carried out through a series of eight measurement steps, of which four include measurements of fermion pairs on two different Majorana islands--hexons and a single tetron--while the other four of these measurement steps are measurements of fermion pairs on a single Majorana island--hexons.

[0054]

[0058] These eight steps in each circuit are preceded by two 2-MZM measurements on the hexon to prepare it to the appropriate initial state in order to measure the stabilizer in the associated surface code plaquette. The eight steps are followed by a final measurement of the hexon to determine the value of the syndrome. Therefore, the total number of measurements to perform stabilizer measurements using either of the exemplary measurement circuits 410 and 412 is eleven, which is believed to be the minimum number of measurements that can be performed in a measurement-only qubit framework for surface code stabilizer measurements.

[0055]

[0059] As mentioned above, any measurement L X Or L Z The maximum number of Majorana islands involved is 2, and the majority of measurements involve only one island. This is advantageous, since the more islands involved in a measurement, the more difficult the corresponding measurement becomes, the lower its quality and the higher its error probability.

[0056] L X and L Z Efficient calculation of measuring circuits

[0060] Such L measurements are shown as measurement circuits 410 and 412 in FIG. X (Equation 11) and L Z To derive the optimized combination of (Eq. 12), we first consider the individual CNOT gates C(X) (h,t) and C(X) (t,h) Find a measurement sequence that compiles separately C(X) in Equations 11 and 12. (h,t) and C(X) (t,h)for each of these sequences, and finally, one may attempt to reduce the sequence length using known methods.

[0057]

[0061] In this case, it is useful to find a measurement sequence compilation by identifying stabilizers and logic operations in a system containing hexons and tetrons, and updating them appropriately as the measurement sequence is performed. If the set of stabilizers at the end of the measurement sequence is the initial set of stabilizers, then the sequence results in a logic gate determined by a transformation of the logical Pauli operators. The logical Pauli operators are C(X) (a,b) If it changes in the same way as it does under conjugation by, i.e.

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[0058]

[0062] As mentioned above in the section “Single Hexon State Space and Operators”, a hexon encodes one logical qubit into six MZMs, with the total parity of the island i 3 The parity sector is stabilized by γ1γ2γ3γ4γ5γ6=+1 and limited to a further auxiliary parity sector, which is, by way of example and not limitation, iγ3γ4=p 34 = ±1. Thus, the set of generators for the initial hexon stabilizer group is S hex = 3 γ1γ2γ3γ4γ5γ6,iγ3γ4>. The corresponding logical Pauli operators (acting on logical qubits) of the hexon island are

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[0059]

[0063] ​Similarly, the Tetron encodes one logical qubit into four MZMs, with the total parity of the island i 2 It is stabilized by γ1γ2γ3γ4. Therefore, the stabilizer group is S tet = 2 γ1γ2γ3γ4>. The corresponding logical Pauli operators are

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[0060]

[0064] Operator Gamma M When measurements are performed, the stabilizers and logical operators are updated according to the rules set out in Table 2.0 below.

[0061] [Table 2]

[0062]

[0065] In considering the stabilizer for the purposes of gate synthesis, the total parity of each island is always fixed (this is violated only by quasiparticle poisoning errors that flip the island parity, which can be ignored for our considerations here, and therefore the total island parity (i for hexons) is 3 γ1γ2γ3γ4γ5γ6=+1, i in Tetron 2 We can assume that the stabilizers corresponding to (γ1γ2γ3γ4=+1) remain implicit. C(X) (h,t) Compilation example of C(X) (h,t) An example of a measurement sequence that realizes X Or L Z The four CNOT operations in the measurement circuit given by (one of the four CNOT operations in the measurement circuit given by) are shown in Table 3.0 below.

[0063] ​ [Table 3]

[0064]

[0066] In Table 3.0, using the shorthand notation ab|cd (as previously introduced for the measurement circuit in Table 2.0),

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[0065]

[0067] The effect of the above measurement sequence is controlled on hexon and targeted to tetron, C(X) (h,t) The gate is applied up to the Pauli operator. X A circuit can be constructed by concatenating variations of this circuit for each of the four tetrons. The efficiency can then be improved by using known sequence manipulation and reduction tools. The same can be done for C(X) (t,h) Gate and L Z It can also be done for circuits.

[0066]

[0068] From the rules in Table 2.0 above, we can see that reversing the measurement sequence gives us the reverse compilation gate C(X). † = C(X), the corresponding measurement sequence can be inverted at will as follows:

number

[0067]

[0069] In particular, the initialization of iγ3γ4 is assumed herein such that all sequences implicitly start with the iγ3γ4 stabilizer.

number

number

number

[0068]

[0070] Next is Full L X The circuit can be summarized and shortened as follows:

number

[0069]

[0071] Here, the first column corresponds to an auxiliary hexon (e.g., hexon 302), and the next four columns correspond to each of the neighboring tetrons (e.g., 306, 312, 316, and 314). This shortens the naive length 16 measurement sequence to a length 8 measurement sequence, where each tetron only involves a single 4-MZM.

[0070]

[0072] Similarly, C(X) (t,h) The gate is L Z They can be used to construct circuits, compiled, combined, and then shortened.

number

[0071]

[0073] L X and L Z For circuit optimization, X and L Z It should be understood that more is involved than minimizing the number of steps in the circuit. In experiments, certain measurements are more difficult to perform than others and therefore involve longer computation times. For example, measurements on MZMs that are closer to each other can be expected to have a lower probability of being faulty and require fewer resources than measurements involving MZMs that are farther apart. According to one approach, available length-8 sequences are identified and an optimal circuit is selected using a cost function that takes into account the relative difficulty of different measurements within each sequence.

[0072]

[0074] One exemplary approach to such sequence optimization is to first X and L Z It is provided that the circuit is divided into two segments, each of which contains two applications of C(X) that can be kept as a pair and shortened. With this in mind, we then alternate between 4-MZM and 2-MZM measurements (each 4-MZM measurement pairs a hexon with a tetron in a different orientation on the lattice, whether it is up, right, left, or down), up to the total Pauli count.

number

number

[0073]

[0075] To find the optimal measurement sequence among these eight possible measurement sequences, a cost function may be used to assign a "difficulty weight" to the particular measurement operations within each sequence. In particular, the measurement difficulty depends on the characteristics of the experimental setup utilized. Thus, the difficulty weight should be selected based on the particular experimental setup selected. By way of example and not of limitation, the following discussion considers the following: X and L Z A methodology for determining and quantifying the difficulty of a particular measurement within a measurement sequence is presented.

[0074]

[0076] The difficulty of performing each measurement may vary depending on several factors, including, for example: In the cutter gate-hexon architecture, measurements are performed by coupling different MZMs to the quantum dot, which effectively forms an interference loop bounded by paths connecting the MZMs through the hexon and paths connecting the MZMs through the dot. To select the interference path, electrostatic depletion gates are adjusted, which effectively connect or disconnect different parts of the semiconductor and define the quantum dot there. These gates, hereafter also called "cutter gates", affect the measurement difficulty in two ways: (i) disorder in the region where the cutter is deposited can locally reduce the phase coherence of the semiconductor and therefore reduce the visibility of the measurement, and (ii) the total length and volume of the semiconducting path can affect the phase coherence as well as properties of the dot such as charging energy and level spacing. In general, measurements may be easier the smaller the dot. Thus, the length of the semiconducting region containing the MZMs affects the difficulty of each measurement, and the number of vertical cutter gates involved in the measurement can be used as a simple placeholder for the length of the semiconducting region. Tunnel junctions - In the described architecture, each coupling between the MZM and the semiconductor can be carefully tuned by a depletion gate forming a tunnel junction. In contrast to cutter gates between semiconducting regions, which are generally either fully open or fully closed, it is important to carefully tune the coupling to the MZM so that its ratio with the charging energy EC is in a favorable regime where the effect on the quantum dot is quickly and reliably measurable without suppressing the charging energy of the dot and increasing the probability of quasiparticle poisoning. In reality, the visibility of the signal may decrease with each tunnel junction and noise at the tunnel gates may affect the measured signal. Thus, the number of tunnel junctions involved in the measurement affects the difficulty of the measurement. Flux noise - In the exemplary architectures discussed above, the energy shift of the quantum dot may depend on the magnetic flux enclosed in the loop. Noise in the enclosed flux, either from noise in the background field or from any magnetic flux lines used to tune the local field, makes the measurement more difficult. Since the magnetic flux noise depends on the enclosed area, this area represents another factor that potentially affects the difficulty of the measurement. Number of islands - In the above architecture, the difficulty of the measurement also depends on the number N of hexons involved. This is because the measurement visibility can be affected by how well one can tune the system to the resonant tunneling point and because the calculations used in the measurement can introduce errors that transfer fermions between different hexons.

[0075]

[0077] By way of example and not limitation, the difficulty weight of a fermion parity measurement of a 2N-MZM M=(jk;l′m′, ) involving N hexons in a system implementing the above architecture may be expressed by the following cost function:

number

[0076]

[0078] Using the cost function above or a similar formula, X and L Z The efficiency of each of the measurement sequences can be assessed. In particular, the tetron / hexon encoding can be represented as a choice of how the physical MZMs are assigned labels (e.g., 1, 2, ...), with the difficulty weights of the various measurements depending on the labeling configuration used. For example, starting from the top left and going counterclockwise, the MZMs in tetrons can be labeled as <1,2,3,4> or <1,3,2,4>. For example, iγ1γ2 (note that

number

[0077]

[0079] According to one embodiment, the next step in the optimization process involves searching through all hexon labeling configurations to find the L with the lowest weight. X Record the sequence and the hexon configuration in which it is realized, L Z This is accompanied by determining the difficulty weight for each sequence by doing the same for C X Labeling, Sequences, and Weights and C Z Given the labeling, sequence, and weights, C X and C Z The Tetron labeling weight can be defined as the geometric mean of

[0078]

[0080] According to one embodiment, this process leads to the identification of an optimized labeling configuration consistent with that of FIG. 4, in which Tetron uses the <1,3,2,4> configuration and M X Hexon uses <5,2,1,3,4,6>, M X Hexon uses the <1,6,2,3,4,5> configuration, and the optimized measurement sequence is the exemplary L X and L Z This is illustrated by circuits 410 and 412.

[0079]

[0081] As mentioned above, the exemplary L shown in FIG. X Circuit 410 and L Z Circuit 412 represents the shortest measurement sequence that can be used to accomplish the four CNOT operations of the surface code stabilizer measurement. Each measurement of these optimized solutions contains at most two Majorana qubit islands—one hexon and one tetron.

[0080]

[0082] FIG. 5 and the following discussion are intended to provide a brief overview of an exemplary computing environment in which the disclosed technology may be implemented. Although not required, the disclosed technology is generally described in the context of computer-executable instructions, such as program modules, executed by a personal computer (PC). Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Furthermore, the disclosed technology may be implemented with other computer system configurations, including handheld devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, network PCs, minicomputers, mainframe computers, etc. The disclosed technology may also be implemented in a distributed computing environment, in which tasks are performed by remote processing devices linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices. Typically, classical computing environments are coupled to quantum computing environments, which are not shown in FIG. 5. 5, an exemplary system implementing the disclosed technology includes a general-purpose computing device in the form of an exemplary conventional PC 500 including one or more processing units 502, a system memory 504, and a system bus 506 coupling various system components including the system memory 504 to the one or more processing units 502. The system bus 506 can be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. The exemplary system memory 504 includes a read only memory (ROM) 508 and a random access memory (RAM) 510. A basic input / output system (BIOS) 512 containing the basic routines that help to transfer information between elements within the PC 500 is stored in the ROM 508.

[0081]

[0083] In one embodiment, system memory 504 stores control logic for the syndrome measurement circuitry and decoding logic 511, such as logic specifically implemented by the QECC and various system decoders.

[0082]

[0084] The exemplary PC 500 further includes one or more storage devices 530, such as a hard disk drive for reading from and writing to a hard disk, a magnetic disk drive for reading from and writing to a removable magnetic disk, and an optical disk drive for reading from and writing to a removable optical disk (such as a CD-ROM or other optical media). Such storage devices may be connected to the system bus 506 by a hard disk drive interface, a magnetic disk drive interface, and an optical drive interface, respectively. The drives and associated computer-readable media provide non-volatile storage of computer-readable instructions, data structures, program modules, and other data for the PC 500. Other types of computer-readable media capable of storing data accessible by a PC, such as magnetic cassettes, flash memories, cards, digital video disks, CDs, DVDs, RAM, ROM, etc. may also be used in the exemplary operating environment.

[0083]

[0085] A number of program modules, including an operating system, one or more application programs, other program modules, and program data, may be stored in the storage device 530. Decoding logic may be stored in the storage device 530 similar to or in addition to the memory 504. A user may enter commands and information into the PC 500 through one or more input devices 540, such as a keyboard and a pointing device such as a mouse. Other input devices may include a digital camera, a microphone, a joystick, a game pad, a satellite dish, a scanner, and the like. These and other input devices are often connected to the one or more processing units 502 through a serial port interface coupled to the system bus 506, but may also be connected by other interfaces such as a parallel port, a game port, or a universal serial bus (USB). A monitor 546 or other type of display device is also connected to the system bus 506 through an interface such as a video adapter. Other peripheral system devices 545, such as speakers and a printer (not shown), may also be included.

[0084]

[0086] The PC 500 may operate in a networked environment using logical connections to one or more remote computers, such as a remote computer 560. In some examples, one or more network or communication connections 550 are included. The remote computer 560 may be another PC, a server, a network PC, or a peer device or other common network node, and typically includes many or all of the elements described above with respect to the PC 500, although only a memory storage device 562 is shown in FIG. 5. The personal computer 500 and / or the remote computer 560 may be connected to logical local area networks (LANs) and wide area networks (WANs). Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets, and the Internet.

[0085]

[0087] When used in a LAN networking environment, the PC 500 is connected to the LAN through a network interface. When used in a WAN networking environment, the PC 500 typically includes a modem or other means for establishing communications over the WAN, such as the Internet. In a networked environment, program modules depicted relative to the personal computer 500, or portions thereof, may be stored in the remote memory storage device or other location on the LAN or WAN. The network connections shown are exemplary and other means of establishing a communications link between the computers may be used.

[0086]

[0088] Example quantum devices disclosed herein include a syndrome measurement circuit that implements an error-correcting code using a plurality of Majorana qubit islands, the syndrome measurement circuit being configured to perform a syndrome measurement by performing a series of measurement-only operations, each one involving at most two Majorana qubit islands.

[0087]

[0089] In one example quantum device of any of the above devices, each of the Majorana quantum islands is either a Majorana tetron or a Majorana hexon.

[0088]

[0090] In another example quantum device of any of the preceding devices, a plurality of Majorana qubit islands are arranged in a rectangular array.

[0089]

[0091] In yet another example quantum device of any of the preceding quantum devices, the error correcting code is a surface code.

[0090]

[0092] In yet another example quantum device of any of the preceding quantum devices, a Majorana qubit island includes tetrons representing data qubits and hexons representing ancillary qubits.

[0091]

[0093] In yet another example quantum device of any preceding device, the syndrome measurements include a plurality of stabilizer measurements, each of which is implemented by a series of connected fermion parity measurements.

[0092]

[0094] In yet another example of any of the above quantum devices, each measurement in a series of linked fermion parity measurements measures either two Majorana zero modes or four Majorana zero modes.

[0093]

[0095] In another example quantum device of any of the preceding devices, a sequence of joint fermion parity measurements is optimized with respect to a measurement resource cost function.

[0094]

[0096] In yet another example quantum device of any of the preceding devices, each of the stabilizer measurements is made by a series of 11 connected fermion parity measurements, where 7 of the 11 fermion parity measurements involve two Majorana zero modes and 4 of the 11 fermion parity measurements involve four Majorana zero modes.

[0095]

[0097] One example method disclosed herein provides for implementing syndrome measurements for an error-correcting code through measurement-only operations by performing a series of measurement-only operations on a plurality of Majorana qubit islands, each one involving at most two Majorana qubit islands.

[0096]

[0098] In one example method according to any of the previous methods, each of the Majorana qubit islands is either a Majorana tetron or a Majorana hexon.

[0097]

[0099] In yet another example method of any preceding method, the plurality of Majorana qubit islands are arranged in a rectangular array.

[0098]

[0100] In yet another example method of any preceding method, the error correcting code is a surface code.

[0099]

[0101] In yet another example method of any preceding method, the Majorana qubit island includes tetrons representing data qubits and hexons representing ancillary qubits.

[0100]

[0102] In yet another example method of any preceding method, performing the syndrome measurements further includes performing a stabilizer measurement on each of a plurality of plaquette in the surface code, the stabilizer measurement being performed by a series of concatenated fermion parity measurements.

[0101]

[0103] In yet another example method of any preceding method, each measurement in the series of linked fermion parity measurements measures either two Majorana zero modes or four Majorana zero modes.

[0102]

[0104] In yet another example method of any preceding method, the sequence of joint fermion parity measurements is optimized with respect to a measurement resource cost function.

[0103]

[0105] In another example method of any preceding method, the stabilizer measurement is performed by a measurement sequence of eleven connected fermion parity measurements, where seven of the eleven fermion parity measurements involve two Majorana zero modes and four of the eleven fermion parity measurements involve four Majorana zero modes.

[0104]

[0106] An example system disclosed herein includes means for performing syndrome measurements of an error-correcting code through measurement-only operations by performing a series of measurement-only operations on a plurality of Majorana qubit islands, each one involving at most two Majorana qubit islands.

[0105]

[0107] Another example quantum device disclosed herein includes a syndrome measurement circuit that performs syndrome measurements of an error-correcting code by performing a series of measurement-only operations on the hexons and tetrons in an array of Majorana qubit islands.

[0106]

[0108] In another example quantum device of any of the above devices, each measurement in a sequence of measurement-only operations involves at most two islands in the array of Majorana qubit islands.

[0107]

[0109] The above specification, examples, and data, together with the attached Appendix A, provide a complete description of the structure and use of the exemplary embodiments. Since many embodiments can be made without departing from the spirit and scope of the invention as set forth in the claims, the following appended claims define the invention. Moreover, structural features of different examples can be combined in yet other embodiments without departing from the scope of the claims as set forth. The above specification, examples, and data provide a complete description of the structure and use of the exemplary embodiments. Since many embodiments can be made without departing from the spirit and scope of the invention as set forth in the claims, the following appended claims define the invention. Moreover, structural features of different examples can be combined in yet other embodiments without departing from the scope of the claims as set forth.

Claims

1. 1. A quantum device comprising: a syndrome measurement circuit that implements an error correction code using a plurality of Majorana qubit islands, the syndrome measurement circuit being configured to perform a syndrome measurement by performing a measurement-only operation, each of a plurality of stabilizer measurements being implemented by a series of concatenated fermion parity measurements, each of the concatenated fermion parity measurements involving at most two of the Majorana qubit islands.

2. 2. The quantum device of claim 1 , wherein each of the Majorana qubit islands is either a Majorana tetron or a Majorana hexon.

3. 10. The quantum device of claim 1 , wherein the plurality of Majorana qubit islands are arranged in a rectangular array.

4. The quantum device of claim 1 , wherein the error correcting code is a surface code.

5. 10. The quantum device of claim 1 , wherein the Majorana qubit island includes tetrons representing data qubits and hexons representing ancillary qubits.

6. 10. The quantum device of claim 1, wherein each measurement in the series of connected fermion parity measurements measures either two Majorana zero modes or four Majorana zero modes.

7. The quantum device of claim 1 , wherein the set of concatenated fermion parity measurements is optimized with respect to a measurement resource cost function.

8. 2. The quantum device of claim 1 , wherein each of the stabilizer measurements is made by a series of eleven connected fermion parity measurements, seven of the eleven fermion parity measurements involving two Majorana zero modes and four of the eleven fermion parity measurements involving four Majorana zero modes.

9. 1. A method for performing an error-correcting code by performing a series of stabilizer measurements, each of the stabilizer measurements being performed by a series of connected fermion parity measurements through measurement-only operations on a plurality of Majorana qubit islands, each of the measurement-only operations involving at most two of the Majorana qubit islands.

10. 10. The method of claim 9, wherein each of the Majorana qubit islands is either a Majorana tetron or a Majorana hexon.

11. 10. The method of claim 9, wherein the plurality of Majorana qubit islands are arranged in a rectangular array.

12. The method of claim 9 , wherein the error correction code is a surface code.

13. 10. The method of claim 9, wherein the Majorana qubit island includes tetrons representing data qubits and hexons representing ancillary qubits.

14. Conducting the syndrome measurement includes: The method of claim 9 , further comprising performing stabilizer measurements on each of a plurality of brackets in the surface code.

15. 15. The method of claim 14, wherein each measurement in the series of connected fermion parity measurements measures either two Majorana zero modes or four Majorana zero modes.

16. The method of claim 14 , wherein the set of concatenated fermion parity measurements is optimized with respect to a measurement resource cost function.

17. 15. The method of claim 14, wherein the stabilizer measurements are performed by a measurement sequence of eleven connected fermion parity measurements, seven of the eleven fermion parity measurements involving two Majorana zero modes and four of the eleven fermion parity measurements involving four Majorana zero modes.

18. 1. A quantum device comprising: a syndrome measurement circuit that performs a syndrome measurement of an error-correcting code by performing a series of measurement-only operations on tetrons and hexons in an array of Majorana qubit islands, the Majorana qubit islands including tetrons representing data qubits and hexons representing ancillary qubits.

19. 20. The quantum device of claim 18, wherein each measurement in the set of measurement-only operations involves at most two islands in the Majorana qubit islands.

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