Majorana-based only-measurement surface-layer encoding architecture

Abstract

A quantum device comprising a syndrome measurement circuit implementing an error-correcting code using a plurality of Majorana quantum bit islands. The syndrome measurement circuit is adapted to implement a syndrome measurement by performing a sequence of only-measure operations, wherein each only-measure operation involves at most two Majorana quantum bit islands.

Description

Technical Field

[0001] This disclosure relates to a quantum device and a calibrator measurement method. Background Technology

[0002] Majorana fermions are their own antiparticle fermions. The zero modes of Majorana fermions can reside on defects, such as domain boundaries or vortices. These Majorana zero modes (MZMs) are attractive for quantum computing because they can carry nonlocal, topologically protected state spaces. Recent experiments have shown that MZMs can exist at the ends of one-dimensional semiconductor lines attached to superconductors. MZMs are currently being explored as potential building blocks for qubits that can be used to perform topological quantum computing.

[0003] One topological quantum computing approach using MZMs employs a scheme in which MZMs move relative to each other, and these moves produce a sequence of transformations (referred to as “weave exchanges” or “weave transformations”) in their nonlocal state spaces. These transformation sequences can represent the corresponding computational gates acting on the qubit states encoded in these state spaces. For example, in a particular encoding scheme, moving one quasi-particle clockwise around another quasi-particle can correspond to a NOT gate (Pauli X) acting on the qubit. To read out the qubit, these quasi-particles might be forced to collide together on the nanowire, allowing the result to be measured. There is evidence that the weave transformations of MZMs can be used to generate Clifford gate sets acting on qubit systems.

[0004] Another topological quantum computing approach using MZMs employs anyon simulations of quantum teleportation to generate weaving transformations without physically transferring the MZMs to one another. This particular approach represents an example of "measurement-only topological quantum computing." In this approach, quantum teleportation is achieved through projective measurements of the joint fermionic parity of paired MZMs and the use of auxiliary MZMs. Repeated applications of quantum teleportation may have the same effect as the weaving transformations of the two MZMs in the aforementioned approach where the MZMs are physically moved. Thus, a sequence of MZM measurements is performed to realize the Clifford quantum gate set. The technique disclosed herein belongs to the latter approach. Summary of the Invention

[0005] According to one implementation, the quantum device includes a collimator measurement circuit that implements error-correcting codes using multiple Majorana qubit islands. This collimator measurement circuit is adapted to perform collimator measurements by executing a sequence of measurement-only operations, each involving up to two Majorana qubit islands. Attached Figure Description

[0006] Figure 1An example quantum computing system is shown that performs error correction via a quantum measurement circuit that uses a Majorana-based measurement-only architecture to implement surface encoding.

[0007] Figure 2A An example bifacial Majorana hexabody architecture is shown that can be used to implement efficient surface-encoded collimator measurements.

[0008] Figure 2B An example single-sided Majorana six-subbody architecture is shown that can be used to implement efficient surface-encoded collimator measurements.

[0009] Figure 3 An example surface coding architecture formed by mesoscopic superconducting islands is shown, which includes Majorana hexapods and Majorana tetrapods.

[0010] Figure 4 Further aspects of an example surface coding architecture including Majorana six-sub-islands and Majorana four-sub-islands are shown.

[0011] Figure 5 The following discussion is intended to provide a brief, general description of an exemplary computing environment in which the disclosed technologies may be implemented. Detailed Implementation

[0012] Quantum computers can perform error correction using quantum error-correcting codes (QECC). QECC encodes logical qubits into a set 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. As long as the physical error rate remains below an acceptable threshold at the cost of increasing the number of physical qubits, QECC enables fault-tolerant quantum computing.

[0013] In quantum processing, the entropy of the data qubit encoding the protected data is transferred to an auxiliary qubit that can be discarded. The auxiliary qubit is positioned to interact with the data qubit so that errors can be detected by measuring the auxiliary qubit and corrected using a decoding unit.

[0014] An important class of error-correcting codes is the stable subcode. In the case of stable subcodes, a logical qubit is defined as a space of simultaneous +1 eigenvectors of a certain number of commuting multi-qubit Pauli operators, called a stable. By repeatedly measuring the quantum system using a complete set of commuting stables, the data bit associated with each auxiliary bit is forced into a simultaneous and unique eigenstate of all stables, thus allowing measurement of stables without disturbing the system. When the measurement result changes, this corresponds to one or more qubit errors, and the quantum state is projected onto different stable eigenstates through measurement.

[0015] Errors are detected by repeated measurements of the stabilizer, and deviations from the expected result of +1 indicate the error.

[0016] Within the category of stable subcodes, surface coding is considered the most promising QECC for fault-tolerant quantum computing. The simplest implementation is defined on a rectangular lattice of qubits, whose squares are divided into two sublattices in a checkerboard pattern. Each square has a stabilizer: for one sublattice, it is given by the product of the data qubits surrounding the square and four Pauli X operators; for the other sublattice, it is given by the product of the corresponding four Pauli Z operators.

[0017] In most practical schemes used to implement surface encoding, the stabilizing qubit measurement (e.g., the product of four Pauli operators performed on the data bits around the square) is achieved by adding an additional auxiliary qubit to each square, entangled with the neighboring data qubit in a special way, and finally performing a single qubit measurement on the auxiliary qubit.

[0018] Since measurements of Pauli operators are transformed into topology-protected parity-check measurements in MZM-based architectures, Pauli stable subcodes ideally map to such architectures. According to one implementation of the disclosed technique, surface encoding is implemented in a quantum device via measurement circuits formed by mesoscopic superconducting islands, which are more specifically referred to herein as Majorana six-subsets (or simply "six-subsets") and Majorana four-subsets (or simply "four-subsets"). In implementing surface encoding to correct measurement errors, optimized stable sub-measurements are achieved by measuring a topology-protected sequence of Majorana-based qubits arranged on nearest-neighbor mesoscopic superconducting islands. According to the proposed method, this stable sub-measurement can be performed using only measurements involving two or four MZMs simultaneously (e.g., 2-MZM or 4-MZM measurements), where each such measurement involves MZMs on a single island or two nearest-neighbor islands. Due to the reduction in the number of measurements and the proximity of the MZMs involved in the measurements, this stable sub-measurement can be implemented with significantly fewer processing resources compared to previous measurement-only methods.

[0019] under, Figure 1 The accompanying description provides an overview of the quantum computing system and the type of measurement circuitry, which can be implemented using the proposed Majorana-based measurement-only surface-encoding architecture. Figure 2 and the accompanying description discuss the various building blocks of this architecture, while regarding... Figures 3-4 The discussion introduces the details of the proposed surface coding architecture and exemplary optimizations of this architecture.

[0020] Figure 1An example of a quantum computing system 100 is illustrated, which performs error correction via a collimator measurement circuit 114 that implements surface encoding using a Majorana-based measurement-only architecture.

[0021] Quantum computing system 100 includes a controller 102 that performs computations by manipulating qubits within a quantum register 108. To enable fault-tolerant quantum computing in quantum computing system 100, readout device 112 includes a calibrator measurement circuit 114 that applies surface-coded calibration to the qubits in quantum register 108. The calibrator measurement circuit 114 uses additional qubits (referred to as “auxiliary qubits”) to perform computations involving the data qubits in quantum register 108. The calibrator measurement circuit 114 performs measurements of the auxiliary qubits in the quantum computer to extract information about the measurements provided regarding errors (faults). To avoid error accumulation during quantum computing, calibrator data is continuously measured, generating r calibrator bits for each round of calibrator measurements. In one implementation, calibrator data is measured at a frequency of 1 µs.

[0022] The calibrator data output by the calibrator measurement circuit 114 is sent to the decoding unit 116, which implements one or more QECCs to analyze the calibrator data and detect the location of each error within the measurement circuit 114 and correct each error.

[0023] By way of example and not limitation, view 142 illustrates a portion of the surface coding architecture for forming the collimator measurement circuit 114. This surface coding architecture encodes a combination of data qubits and auxiliary qubits, the data qubits being represented by hollow circles (e.g., data qubit 118) and the auxiliary qubits by solid circles (e.g., auxiliary qubit 120). The data qubits and auxiliary qubits are arranged in a crystal-like structure defined by multiple squares (e.g., squares 122, 124), each square comprising a central auxiliary qubit and four data qubits arranged at its corners.

[0024] In the quantum processing, the entropy from the qubits encoding protected data is transferred to auxiliary qubits (e.g., auxiliary qubits from the corners of the square to the center), which can be discarded after the measurement. Data transfer between the data qubits at the corners of each square and the auxiliary qubits at the center (e.g., auxiliary qubit 120) is achieved via a process that involves winding the data qubits at the four corners and the auxiliary qubit at the center together, and then performing a projection measurement on the central auxiliary qubit. This process, referred to herein as a "stabilizer measurement," produces a single correction qubit, which is then sent to the decoding unit 116.

[0025] The surface encoding enables two types of stabilizer measurements—X-type stabilizer measurements and Z-type stabilizer measurements. Each square in the calibrator measurement circuit 114 is used to perform either an X-type or Z-type stabilizer measurement. For example, a shaded square (e.g., square 124) can be used for a Z-type stabilizer measurement, and an unshaded square can be used for an X-type stabilizer measurement. In this architecture, the auxiliary qubit at the center of each shaded square is called a “measurement-Z” qubit, while the auxiliary qubit at the center of each unshaded square is called a “measurement-X” qubit. Each measurement-X and measurement-Z qubit is coupled to four data qubits (e.g., four nearest hollow circles), and each data qubit is coupled to two measurement-Z qubits and two measurement-X qubits (e.g., four nearest solid circles).

[0026] The stable sub-measurement for a single square 122 of surface encoding requires a series of measurements, which at least include: (1) performing a first measurement to initialize the central auxiliary qubit to its computational state |0>; (2) four CNOT operations to wrap all five qubits located on square 122; and (3) a projection measurement of the central auxiliary qubit.

[0027] On the block used to perform Z-type stabilizer measurements, the 4-CNOT measurement (step (2) above) targets a measurement qubit that has four nearest-neighbor data qubits as control, and has the ability to generate The characteristic state is projected and measured. For the block used to perform X-type stabilizer measurements, four CNOTs target the nearest-neighbor data qubit, which uses the measurement qubit as control, and the sequence also includes Hadamard gates applied to the measurement qubit before and after the CNOTs. As an alternative to applying the Hadamard gate in this way, one could use... The operator is used to initialize the measurement of the -X qubit and by measuring the -X qubit Operator measurements are used to perform the final corrector readout (after the CNOT sequence). Projection measurements are generated. The characteristic state. In one implementation where the crystal structure shown represents a single logic qubit, the projected measurements of all measurement qubits in the logic qubit ensure that the state |ψ> of all data qubits in the logic qubit simultaneously satisfies as well as Having eigenvalues Having eigenvalues The following test repeats this cycle. Stabilizer measurements for all auxiliary qubits are performed so that each step in the X-type or Z-type corrector measurement circuit is completed on each square in the crystal before the next step begins.

[0028] In different quantum platforms, the four CNOT operations of surface-encoded stabilizer measurements can be implemented in various ways. In the architecture proposed in this paper, MZMs are used to implement surface encoding. The auxiliary qubits at the center of each square (e.g., square 122) are implemented using an MZM structure called a Majorana six-sub, referred to as "six-sub" in this paper, which consists of six MZMs. The data qubits at each corner of the square are implemented using a structure called a Majorana four-sub, referred to as "four-sub" in this paper, each consisting of four MZMs. For example, Figure 1 View 130 illustrates the Majorana Islands 148, comprising a central hexapod 134 surrounded by four quadruplets 136, 138, 140, and 144. The hexapod 134, together with the quadruplets 136, 138, 140, and 144, forms five data bits on square 122. Surface-encoded stabilizer measurements can be performed by executing a sequence of joint fermion parity measurements involving a small number of MZMs within the Majorana Islands 148. For example, the four CNOT operations of stabilizer measurements can be performed using a sequence of 2-MZM and 4-MZM parity measurements. This measurement-only approach using hexapods and quadruplets allows for more efficient computation than previous methods.

[0029] Figure 2A and 2B Examples of MZM hexapod architectures 200 and 202 are illustrated, which can be used to implement efficient surface-encoded collimator measurements. An MZM hexapod (referred to herein as a "hexapod") is a superconducting island containing six MZMs (e.g., MZM204), some of which are used to encode qubit states, and others are used as auxiliary degrees of freedom to facilitate measurement-based operations. Figure 2A The MZM six-substructure architecture 200 is a double-sided six-substructure, while Figure 2B The MZM six-substructure architecture 202 is a single-sided six-substructure.

[0030] While qubits can also be formed from four MZMs (referred to as quadruplets on isolated superconducting islands), such qubits themselves do not allow any topologically protected unitary gate operations due to the lack of auxiliary MZMs. On the other hand, six-sub-qubits allow for the realization of a complete single-qubit Clifford gate set using topological protection.

[0031] According to one implementation, the MZMs (e.g., MZM 204, MZM 206) included in each of the six sub-body are coupled via a topological superconducting line (e.g., topological superconductor 208) and a spine made of a conventional (S-wave) superconductor (e.g., superconductor 210, superconductor 214).

[0032] exist Figure 2A In the bifacial architecture, three superconducting lines are connected by a central ridge, and MZMs appear at both ends of each superconductor 208. Figure 2B In the single-sided architecture, six nanowires are connected at one end by ridge 210, with MZMs appearing only at the other end. A significant advantage of these architectures is that individual qubit islands are electrically isolated (except for weak coupling with points, see below), thus the Coulomb interaction generates a finite charging energy E for the island. c This helps prevent (external) quasi-particle poisoning because the probability of electrons tunneling into or away from the island from the outside is reduced by the charging energy E. c The ratio to temperature is suppressed exponentially, exp(-E C / k B T). Due to the thermally excited quasiparticles on the island, the decoherence of the topologically protected state is exp(-Δ / k). B T) suppression, where Δ is the topological gap. Due to the virtual tunneling of fermions between MZMs, degeneralization splitting is suppressed by exp(-L / ξ), where L is the spacing between MZMs and ξ is the superconducting coherence length.

[0033] According to one implementation, the projection measurement of the joint fermionic parity of any two MZMs (2-MZM measurements) can be achieved by enabling a somewhat coherent single-electron tunnel between the MZM and adjacent quantum dots (e.g., quantum dot 214, quantum dot 216), forming an interference loop. The projection measurement of the collective fermionic parity of 2N-MZMs can be performed similarly, but care must be taken to ensure that the interference loop involves all 2N MZMs; for example, fermions cannot pass directly between the various quantum dots involved. These couplings cause shifts in the energy spectrum and charge occupancy of the points, depending on the fermionic parity of the MZMs. These shifts, in turn, can be measured using existing techniques developed for charge and spin qubits, such as capacitive induction or quantum capacitance measurements. Importantly, the measurement is topologically protected in a sense, as the operator being measured is known to depend on a correction value that is exponentially small over the distance the MZMs are separated by a superconducting region (nanowire or ridge).

[0034] Multiple six-subs can be arranged in an array, and multi-qubit operations (e.g., CNOT operations used in surface-encoded stabilizer measurements) can be performed by weakly coupling MZMs from different islands (e.g., different six-subs or four-subs) to a common quantum dot. Because the coupling between the MZMs and the quantum dot is weak, charge energy protection against quasi-particle poisoning remains effective during such operations. This limits the operators that can be measured to those that commutate using charge energy (or total parity) on each island, which is precisely the measurement involving an even number of Majorana operators on each island.

[0035] Figure 3 The illustration shows an example surface coding architecture 300 formed by two different forms of mesoscopic superconducting islands: quadruplets and hexapods, all arranged in a rectangular array. While hexapods (e.g., hexapods 302, hexapods 304) can have the same characteristics as described above... Figure 2A and Figure 2B Those identical or similar individual structures discussed, the quadruplets (e.g., quadruplets 306, 308, 310, 312, 314, and 316) can be understood as structures comprising four MZMs, each located at the end of a topological superconductor and attached to the same superconductor.

[0036] Each of the six Majorana sub-quantum units carries six Majorana zero modes, which is the minimum number of MZMs to encode a combination of two qubits. Using one qubit as the data qubit and the other as the auxiliary qubit, arbitrary single-qubit Clifford operations can be performed on the data qubit in a topology-protected manner, with only measurements taken.

[0037] In contrast, each Majorana quad carries four Majorana zero modes, represented by circles in the corners. These four Majorana zero modes can be used to collectively encode a single computational qubit in the nonlocal (topological) state space of an MZM—the joint fermion parity check of the MZM pair.

[0038] In the surface-encoded architecture 300, the quadruplets act as data qubits, while the hexruplets are used as auxiliary qubits to support unitary operations and the implementation of X and Z stable sub-measures. To avoid confusion between the term "auxiliary qubit" referring to the second qubit encoded within the hexruplets and the qubit used to support stable sub-measures in the surface encoding, the auxiliary qubits of the surface encoding are referred to as "auxiliary hexruplets" in the following description.

[0039] exist Figure 3 In the diagram, the auxiliary six-sub-units, represented by light gray shaded squares, correspond to the auxiliary qubits supporting the measurement of the X-stabilizer in the surface encoding. These six-sub-units may also be referred to as "M" in this paper.X - Six-sub-units". Similarly, the auxiliary six-sub-units, represented by dark gray shaded squares, correspond to the auxiliary qubits supporting Z-stableton measurements in the surface encoding. These six-sub-units are referred to in this paper as "M". z -Six-sub-body".

[0040] Such as about Figure 1 The X-stabilizer measurement and Z-stabilizer measurement discussed are implemented via a calibrator measurement circuit that initializes the central auxiliary qubit in its computational state |0>, followed by four CNOT operations and a projection measurement.

[0041] Due to the properties of Majorana hexets and Majorana tetras described above, each CNOT operation in the CNOT operation can be implemented using three Majorana tetras, two Majorana hexets, or (in the proposed method) one Majorana hexet and one Majorana tetra. This hybrid approach is advantageous because it provides an optimized trade-off between efficiency and qubit quality. This is partly due to the longer lifetime (better quality) exhibited by Majorana tetras, but more Majorana tetras are required to perform the same quantum operations that can be performed by a single Majorana hexet. Compared to existing methods, the proposed hexet / tetra hybrid architecture reduces the total number of measurement operations required to implement surface-encoded comparator measurements, while also providing higher overall qubit quality.

[0042] According to one implementation, a surface-encoded collimator measurement circuit is performed for each auxiliary six-subbody, and a sequence of joint fermion parity checks is performed on the nearest neighbor Majorana qubit islands, wherein each individual measurement in the sequence involves no more than two nearest neighbor islands.

[0043] Before introducing specific exemplary measurement sequences for optimizing the efficiency of the surface-encoded corrector measurement circuit, a discussion of the six-subspace state, operators, and symbols is first introduced below.

[0044] Single six-subbody state space and operators

[0045] The MZMs in each six-subunit can be numerically designated as 1, 2, 3, 4, 5, and 6. Similarly, the MZMs in each four-subunit can be numerically designated as 1, 2, 3, and 4. In the notation used in the following discussion, each of these MZMs is represented by the Majorana fermion operator γ. j This represents the MZM at position j. These operators follow the usual fermion anticommutation relation {γ j γ k}=2δ jk For any ordered MZM pair j and k, their joint fermion parity check operator is given by iγ. jγ k =-iγ k γ j Given that the operator has eigenvalues ​​p for odd parity and even parity respectively. jk = ±1. (The convention in this paper is slightly different from that in reference

[15] .) Parity check s = p jk The corresponding projection operator of ±1 on the subspace is given by the following equation:

[0046]

[0047] Then, the operator iγ j γ k It can be represented as

[0048]

[0049] The abbreviation ± for ±1 is used for even parity (vacuum) and odd parity (fermion) channels, respectively.

[0050] In this way, regarding certain choices of fermion parity checks for how to pair fermion parity checks together, the fundamental state p 12 p 34 p 56 This can be used to represent a system with six MZMs. Due to the limited charging energy of the islands, the system generally only has a ground state in the even or odd collective fermion parity region, which can be tuned using the gate voltage; without loss of generality, it can be assumed that the system is tuned to a ground state with even collective fermion parity, i.e., p 12 p 34 p 56 = +1, and the state with an odd number of collective fermions parity is the excited state associated with quasiparticle poisoning. Thus, the low-energy state space of the six-subject is four-dimensional, with the following fundamental states:

[0051] |+,+,+> (3)

[0052] |-, +, ->=iγ2γ5|+, +, +> (4)

[0053] |+,-,->=iγ4γ5|+,+,+>. (5)

[0054] |-,-,+>=iγ2γ3|+,+,+>. (6)

[0055] Consider it as a two-qubit system, where the first qubit is encoded in p 34 And the second qubit is encoded in p 12In this context, the basic states are 0, 0, 0, 1, 1, 0, 1, 1. Then, the MZM parity check operator can be expressed based on the Pauli operators on these two qubits.

[0056]

[0057] The Pauli matrix is:

[0058]

[0059] Figure 3 The numbering convention for the six-subsystem 316 is introduced, in which MZM 3 and MZM 4 are used as auxiliary MZMs with a defined joint parity check, for example, p 34 =+1, and the computational qubit is encoded in p 12 In the middle. The remaining parity checks are related to the other two, such as p. 56 =p 12 p 34 Therefore, when the auxiliary pair has p 34 When =+1, the basic state is calculated as follows:

[0060] |0>=|p 12 =-p 56 =+>,|1>=p 12 =p 56 =-> (9)

[0061] And when p 34 When = -1, the basic state is calculated as follows:

[0062] |0>=|p 12 =-p 56 =+>,|1>=p 12 =-p 56 =->。 (10)

[0063] Another view is that the six-subset is a Majorana stable subcode, which encodes a single logical qubit into six MZMs. In this language, a logical qubit is defined as a set of operators simultaneously +1 in a characteristic space called a stable subset. The logic gates operating on this space are operators that commute with the stable subset but are not themselves stable. For the example of the six-subset, the stable subset is formed by island i 3 The total parity check of γ1γ2γ3γ4γ5γ6 and the parity check generation of auxiliary pair iγ3γ4. Logical Pauli operators are derived from... and

[0064] Corrector measurement circuit for auxiliary six-subbody

[0065] According to one implementation, for a single M XThe collimator measurement circuit for the six-sub-302 (e.g., to measure its logic qubits) requires a sequence of the following operations:

[0066] 1. Place M X - The six-sub-unit 302 is initialized to the state X = +1. The auxiliary qubits of the six-sub-unit are in arbitrary but deterministic states (e.g., iγ1γ6 = +, iγ3γ4 = p). 34 state);

[0067] 2. Sequences applying the CNOT operation:

[0068] Controlled in M X -Six sub-body (labeled as h) X On ), and with the four nearest neighbor quadruplets (labeled t) j ) as the target; and

[0069] 3. Measure M on the X basis. X -Six sub-body 302. (For example, measuring iγ1γ6).

[0070] The effect of this series of steps is the four data quadruplets 306, 312, 314, and 316. The final measurement result (step 3) is the result of the stabilizer measurement.

[0071] According to one implementation, for a single M z The calibration measurement circuit of the six-subbody 304 requires the following sequence of operations:

[0072] 1. Place M X - The six-sub-qubit is initialized to state 0 (Z = +1), and the auxiliary qubits of the six-sub-qubit are in arbitrary but deterministic states (i.e., iγ1γ2 = +, iγ3γ4 = p). 34 state);

[0073] 2. Sequences using CNOT:

[0074] Controlled within four nearest-neighbor quadruplets (labeled t) j On, and with M Z Six-subbody (labeled as h) z ) as the target; and

[0075] 3. Measure M on the Z-base. Z - Six sub-sub ...

[0076] The effect of this series of steps is four data quadrilaterals. The final measurement result (step 3) is the result of the stabilizer measurement.

[0077] The most efficient computations described above depend on the optimized compilation of these circuits (e.g., steps 1-3 in each of the scenarios above). Since steps 1 and 3 are simply measurements (two are needed for step 1 and one for step 2), there is no room for optimization. Therefore, instead, we can focus on step 2 and search for the optimal measurement sequences for the two sequences that implement the CNOT gate, which can be represented as:

[0078]

[0079] Figure 4 It shows about L X Measurement circuit 410 and L Z Further details of the exemplary surface coding architecture 400, an exemplary representation of measurement circuit 412 (see Equations 11 and 12 above), which can be used to implement four CNOTs in each of the X-type and Z-type stabilizing sub-measurement circuits, are provided below with reference to exploded diagram 402 of the surface coding architecture 400 and an exemplary MZM numbering scheme. Figure 4 L X and L Z The symbols required for the circuit.

[0080] Exploded Figure 402 illustrates a portion of the surface coding, comprising six quadruplets (labeled with the letters A, B, C, D, E, F, G) and two hexadecimals (M). X and (M) z In this structure, M X The six sub-units are surrounded by four sub-units A, B, C, and D, each of which acts as a single data qubit. M z The six-sub-unit is similarly surrounded by four four-sub-units E, F, A, and B, which act as data qubits. Within each six-sub-unit and four-sub-unit, the MZM is labeled by numbers according to an exemplary numbering scheme. The Majorana four-sub-units each support modes 1, 2, 3, and 4, while the Majorana six-sub-units support modes 1, 2, 3, 4, 5, and 6.

[0081] Figure 4 Measurement circuits 410 and 412 each provide a sequence of projection measurements for 2-MZM and 4-MZM joint fermion parity checks, which, according to an optimization method, can be executed to achieve the L proposed in equations 11 and 12 above (e.g., a sequence of four CNOTs in the X and Z type corrector measurement circuits). X and L Z Specifically, the measurement circuit 410 is illustrated for measuring the mass of Majorana tetramers A (M). XThe sequence of projection measurements of auxiliary qubits encoded by the Majorana quadruplets (M-6-sub-units) is illustrated in measurement circuit 412, which is used to measure the sequence of projection measurements of auxiliary qubits encoded by the Majorana quadruplets (M-6-sub-units). Z The sequence of projection measurements of auxiliary qubits encoded by (-six-sub-units).

[0082] The specific fermion parity measurements and measurement order shown in measurement circuits 410 and 412 are intended to be exemplary but not exclusive. Clearly, there are countless L-type methods for implementing a single measurement with variable sequence length (number of measurements) and variable difficulty cost (discussed below). X and L Z The solution is as follows. Measurement circuits 410 and 412, also shown in Table 1.0 below, represent an optimized solution derived from the discussion below.

[0083] Table 1.0

[0084]

[0085] L X and L Z Measurement circuit symbol

[0086] Reference measurement circuit 410(L) X (Also shown in Table 1.0 above) The first column corresponds to M X The sixth column corresponds to M in the second column. Z The six sub-body structures are shown in six columns, and the following four columns, from left to right, correspond to the four sub-body structures (A, B, C, D, E, and F) positioned as shown in exploded diagram 402. Eight measurement steps are shown in different rows from top to bottom. These steps are used to achieve L. X And thus realize the auxiliary six-sub-unit (A) (representing the auxiliary qubit in a single surface coding square) and its four neighbors (B, C, D and E) (representing the data qubit in a single surface coding square).

[0087] Exemplary, and not limiting, L X The first measurement step in (the top row of the measurement circuit 410) is M. X The 4-MZM measurement of the joint fermion parity check of the six sub-sub ... X 2-MZM measurement of joint fermion parity check of six sub-subsidiaries MZM 1 and MZM 2. The third measurement step (third line) requires M... XThe 4-MZM measurement is a joint fermion parity check of the MZM1 and MZM3 of the six sub-sub ...

[0088] In the measurement circuit 412(L) z In the above, the symbol is the same as that for the measurement circuit 410 (L) X The signs are consistent. L X and L z Each measurement circuit in the measurement circuit (e.g., the four CNOT operations of the stabilizer) is implemented via a sequence of eight measurement steps. Of these eight measurement steps, four involve measurements of fermion pairs on two different Majorana islands—a six-subparticle and a single four-subparticle, while the other four measurement steps involve measurements of a single Majorana island fermion pair—a six-subparticle.

[0089] To measure the stabilizer in the associated surface-encoded lattice, two 2-MZM measurements are performed on the six-subbody before these eight steps in each circuit to bring it to a proper initial state. Following these eight steps is the final measurement of the six-subbody to determine the value of the compensator. Therefore, the total number of measurements performed using either the exemplary measurement circuit 410 or measurement circuit 412 is 11, which is considered the minimum number of measurements that can be achieved for surface-encoded stabilizers within a qubit-only measurement framework.

[0090] As discussed above, any measurement L X or L z The maximum number of Majorana islands involved in the survey is two, and most surveys involve only a single island. This is advantageous because a larger number of islands involved in the survey will lead to greater difficulty, poorer quality, and a higher error rate.

[0091] L X and L z Efficient calculation of measurement circuit

[0092] In order to obtain L X (Equation 11) and L z Optimized compilation of (Equation 12), for example Figure 4 The measurement circuits 410 and 412 shown in the diagram can be used to first find and compile each CNOT gate C(X) individually. (h,t) And C(X) (t,h) The measurement sequences are then inserted into C(X) of Equation 11. (h,t) And C(X) in Equation 12 (t,h) In the process, and finally attempted to reduce the length of the sequence using known methods.

[0093] In this context, identifying the stabilizers and logic operators in the system, which includes both six-subset and four-subset systems, and appropriately updating them as the measurement sequence is executed, is helpful in finding the compileability of the measurement sequence. If the stable subset at the end of the measurement sequence is the same as the initial stable subset, then the sequence will produce a logic gate determined by transformations of the logical Pauli operators. The given measurement sequence will be compiled to the target gate C(X). (a,b) If the logical Pauli operator transformation is the same as that by C(X)... (a,b) The transformations under conjugate are the same, that is...

[0094]

[0095] As discussed in the section "State Space and Operators of a Single Six-Subbody", a six-subbody encodes a logical qubit in six MZMs, by island i 3 The total parity check of γ1γ2γ3γ4γ5γ6=+1 is used for stabilization and is restricted to a further auxiliary parity check region, which is exemplarily, but not restrictively, initialized as iγ3γ4=p 34 = ±1. Therefore, the generator set used to initialize the six-subgroup stable subgroup is S. hex = 3 γ1γ2γ3γ4γ5γ6,iγ3γ4>. The corresponding logical Pauli operator (operating on logical qubits) used for the six-sub-island is and The equivalence class contains all parity operators related to the stabilizer. The 2-MZM parity check for the six-subbody can be mapped back to the Pauli operator via Equation 7 above.

[0096] Similarly, the four sub-body encodes a logical qubit in four MZMs, by island i 2 The total parity check of γ1γ2γ3γ4 is used for stabilization. Therefore, the stable subgroup is S. tet = 2 γ1γ2γ3γ4>. The corresponding logical Pauli operator is and The 2-MZM parity check operator for quadruplets can be mapped back to the Bubble operator as follows:

[0097]

[0098] When the operator г M When the measurement is performed, the stabilizer and logical operator are updated according to the rules presented in Table 2.0 below.

[0099] Table 2.0

[0100] ​​

[0101]

[0102] When discussing the stabilizers used for gate synthesis, it can be assumed that the total parity of each island is always fixed (only the quasi-particle poisoning error violates this, as it flips the island's parity, and can be ignored in this discussion, thus relating to the total island parity (for the i-th sub-island). 3 γ1γ2γ3γ4γ5γ6=+1 and i for the four sub-bodies 2 The corresponding stabilizers (γ1γ2γ3γ4) will remain implicit.

[0103] C(X) (h,t) Example compilation

[0104] Implement C(X) (h,t) Examples of measurement sequences (e.g., those in equations 11 and 12 above, by L) X and L z The following table 3.0 shows one of the four CNOT operators in the given measurement circuit.

[0105] Table 3.0

[0106]

[0107] In Table 3.0, the abbreviation ab|cd (also described above for the measurement circuit in Table 2.0) is used to represent... and The symbols used to indicate that the corresponding six-subgroup or four-subgroup is not involved. As mentioned earlier, the parity stabilizers of the total island are kept implicitly because they are assumed to be fixed throughout the process. Furthermore, the symbols in the stabilizers or logical operators are not explicitly stated. For example, (iγ1γ2)(iγ1γ3) = -iγ2γ3, but will be recorded as 23. The effect of these symbols is to change the compilation gate through the total Pauli operator, which can be determined by Pauli tracing.

[0108] The effect of the above measurement sequence depends on the Pauli operator, applied to C(X) which is controlled on six sub-sub ... (h,t) Door. A complete L X The circuit can be constructed by varying the connections used for each of the four sub-sub ... (t,h) Door and L z Circuits can also be done this way.

[0109] According to the rules in Table 2.0 above, it is known that reversing the measurement sequence produces the inverse of the compiler gate. Since... The corresponding measurement sequence can be freely reversed, as shown below:

[0110]

[0111] It is worth noting that the initialization of iγ3γ4 assumed in this paper causes all sequences to implicitly begin with the iγ3γ4 stabilizer. Because Instantaneous repetition of the same measurement can be reduced. Furthermore, triple measurements of M1 and M2, and then M1, due to the nature of this measurement... in It can be reduced.

[0112] Then the complete L X Circuits can be compiled and reduced in the following ways.

[0113]

[0114] Here, the first column corresponds to the auxiliary six sub-body (e.g., six sub-body 302), and the next four columns correspond to each of the adjacent four sub-body (e.g., 306, 312, 316, and 314). This reduces the original measurement sequence of length 16 to a measurement sequence of length 8, where each four sub-body involves only a single 4-MZM measurement.

[0115] It was used in a similar way to construct L Z The circuit's C(X) (t,h) Gates can be compiled, combined, and then reduced:

[0116]

[0117] This also reduces the original 16-length measurement sequence to an 8-length sequence, where each quadron involves only one 4-MZM measurement. This only indicates that it can be used to achieve L X and L Z One of many different sequences of circuits. It is worth noting that the above-derived sequence for L... X and L Z The solution is achieved through far fewer measurements than other existing measurement-only methods.

[0118] It should be understood that L X and L Z Circuit optimization requires more than just minimizing L. X and L ZThe number of steps in the circuit. Experimentally, some measurements will be more difficult to perform than others and therefore require more computation time. For example, measurements on MZMs that are close to each other can be expected to have fewer failures and require fewer resources compared to measurements involving distant MZMs. According to one approach, sequences of length 8 are available and identifiable, and the optimal circuit is selected using a cost function to account for the relative difficulty of different measurements within each sequence.

[0119] An exemplary method for this sequence optimization provides that first L... X and L Z The circuit is divided into two segments, each involving two C(X) applications, which can be operated and reduced as a pair. With this in mind, it is then possible to search for all measurement sequences of length 4 alternating between 4-MZM and 2-MZM measurements (each 4-MZM measurement pairs a six-sub-body with a four-sub-body in different orientations—up, right, left, or down—on the lattice), and, depending on the total Pauli factor, compile them into... and For each measurement step, there are 8 possible MZM pairs that can be selected as six sub-bodies, and 4 MZM pairs for the selected four sub-bodies. The search space for the 4-MZM, 2-MZM, 4-MZM, 2-MZM measurement sequences with the constraint that the final 2-MZM measurement is on the iγ3γ4 of the six sub-bodies is therefore greater than (8×6)×(7×24+1×48)=10,368 measurement-only sequences. For each pair j and k in the direction, there are 64 sequences for C(X). (h，t) The sequence, and for C(X). (t，h) Similarly. These sequences can then be combined to form L. X and L Z Only measurement compilation. This will generate optimized... and All L obtained from compilation X and L Z List of circuits. Not yet searched for all measurement sequences of length 8 that alternate between 4-MZM and 2-MZM measurements; in this case, the search space has more than (48×8)2×48×9×24×1=1,528,823,808 measurement-only sequences.

[0120] To find the optimal one among these eight possible measurement sequences, a cost function can be used to assign a "difficulty weight" to specific measurement operations within each sequence. It is worth noting that the measurement difficulty depends on the characteristics of the experimental setup utilized. Therefore, the difficulty weight will be selected based on the specific experimental setting chosen. By way of example, and not limitation, the following discussion conveys a method for determining and numerically representing a given L.X and L Z A method for measuring the difficulty of a specific measurement in a sequence.

[0121] The difficulty of achieving a single measurement can vary depending on many factors, such as the following:

[0122] Cut gates—In a six-subbody architecture, measurements are performed by coupling different MZMs to quantum dots, which effectively form interference loops characterized by paths connecting the MZMs through the six-subbody and paths connecting the MZMs through the dots. To select the interference paths, electrostatic loss gates are tuned, which effectively connect or disconnect different parts of the semiconductor and define the quantum dots within them. These gates, hereinafter also referred to as “cut gates,” affect measurement difficulty in two ways: (i) the disorder in the region where the cut is placed may locally reduce the phase coherence of the semiconductor and thus reduce the visibility of the measurement; and (ii) the total length and volume of the semiconductor paths may affect the phase coherence and properties of the dots, such as their charging energy and level spacing. Generally, measurements may be easier for smaller dots. Therefore, the length of the semiconductor region including the MZMs affects the difficulty of each measurement, and the number of vertical cut gates involved in the measurement can be used as a simple placeholder for the length of the semiconductor region.

[0123] Tunnel junctions—In the described architecture, each coupling between the MZM and the semiconductor can be carefully tuned via loss gates that form tunnel junctions. Unlike cut gates between semiconductor regions, which are typically fully open or closed, it is important to carefully tune the coupling with the MZM to achieve a favorable ratio to the charge energy EC, in which the quantum dot can be measured quickly and reliably without suppressing the dot's charge energy or increasing the probability of quasiparticle poisoning. In practice, each tunnel junction can potentially reduce signal visibility, and noise in the tunnel gates can affect the measurement signal. Therefore, the number of tunnel junctions involved in the measurement influences the difficulty of the measurement.

[0124] Magnetic flux noise—In the exemplary architectures discussed above, the energy shift of the quantum dot may depend on the closed magnetic flux within the loop. Noise in the closed magnetic flux, whether it is noise in the background field or any magnetic flux lines used to tune the local field, will make measurements more challenging. Since magnetic flux noise depends on the closed region, this region represents another factor that can affect the difficulty of measurement.

[0125] The number of islands—In the architecture above, the difficulty of measurement also depends on the number of six sub-islands N involved. This is because measurement visibility can be affected by the degree to which the system is tuned to the resonant tunneling point, and also because the operations used in the measurement can cause errors in the transmission of fermions between different six sub-islands.

[0126] By way of example, but not limitation, the difficulty weights of the fermion parity measurement of 2N-NZMs M-(jk; l'm'; ...) involve N six-sub-bodies in the system, and the implementation of the above architecture can be represented by the following cost function:

[0127]

[0128] Where, n c n is the number of vertical doors opened for measurement. t This is the number of tunnel junctions involved in the measurement, which is equal to the number of MZMs involved in the measurement (including the number of MZMs in the coherent link), and n a It is the integer number of unit areas enclosed by the interference rings depicted by the measurements. Number w c w t and w a It is the difficulty weight associated with the corresponding factors mentioned above.

[0129] Using the cost function described above or a similar expression, each L of length 8 can be evaluated. X and L Z The efficiency of the measurement sequence. It is worth noting that the encoding for quad-sub-body / six-sub-body can represent the choice of how to assign tags to physical MZMs (e.g., 1, 2, ...), and the difficulty weights of various measurements depend on the tag configuration used. For example, rotating counter-clockwise from the top left corner, the MZMs on the quad-sub-body can be labeled <1,2,3,4> or <1,3,2,4>. For example, due to the need for coherent links, in the first encoding scheme, iγ1γ2 (which is still related to...) The operator-associated measurement has a lower difficulty weight than the second encoding scheme.

[0130] According to one implementation, the next step in this optimization process requires searching all six-subbody label configurations by recording the lowest weight L. X The sequence determines the difficulty weight of each sequence (e.g., according to the cost function in Equation 16 above) and implements the six-sub-body configuration for that sequence, and for L Z Similarly. For each quadruple tag configuration, this gives C... X Labels, sequences, and weights, as well as C Z Labels, sequences, and weights. The label weight of a quadruplet is defined as its C. X and C Z The geometric mean of the weights can be used to select the optimal label configuration.

[0131] According to one implementation, this process leads to... Figure 4 The matching optimized flag configuration identifier, where the four sub-bodies use the <1,3,2,4> configuration, MX The six sub-units use the <5,2,1,3,4,6> configuration and M X The six sub-units use a <1,6,2,3,4,5> configuration, and the optimized measurement sequence is... Figure 4 Example L X Circuit 410 and L Z The circuit is shown in Figure 412.

[0132] As mentioned above, Figure 4 The exemplary L shown X Circuit 410 and L Z Circuit 412 represents the shortest sequence of four CNOT operations that can be used to implement surface-encoded stabilizer measurements. The single measurement of these optimized solutions involves at most two Majorana qubit islands in a Majorana qubit island, i.e., one quadruple island and one hexaple island.

[0133] Figure 5 The following discussion is intended to provide an outline, a brief, general description of an exemplary computing environment implementing the disclosed techniques. While not strictly required, the disclosed techniques are described in the general context of computer-executable instructions (e.g., program modules) executed by a personal computer (PC). Typically, program modules include routines, programs, objects, components, data structures, etc., which perform a specific task or implement a specific abstract data type. Furthermore, the disclosed techniques can be implemented using other computer system configurations, including handheld devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, network PCs, minicomputers, mainframes, and so on. The disclosed techniques can also be practiced in distributed computing environments, where tasks are performed by remote processing devices linked via a communication network. In distributed computing environments, program modules can reside in both local and remote memory storage devices. Typically, classical computing environments are coupled to quantum computing environments, but… Figure 5 The quantum computing environment is not shown. (Reference) Figure 5 An exemplary system for implementing the disclosed technology includes an exemplary general-purpose computing device in the form of a conventional PC 500, which includes one or more processing units 502, system memory 504, and a system bus 506 that couples various system components, including the system memory 504, to the one or more processing units 502. The system bus 506 can be any of a variety 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 read-only memory (ROM) 508 and random access memory (RAM) 510. A basic input / output system (BIOS) 512 is stored in the ROM 508, containing basic routines that facilitate the transfer of information between components within the PC 500.

[0134] In one implementation, system memory 504 stores control logic for the calibration sub-measurement circuit, as well as decoding logic 511, such as QECC, and logic specifically implemented by various system decoders.

[0135] 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 disk drive for reading from or writing to a removable disk, and an optical disc drive for reading from or writing to a removable optical disc (e.g., a CD-ROM or other optical media). Such storage devices can be connected to the system bus 506 via a hard disk drive interface, a disk drive interface, and an optical disc drive interface, respectively. The drives and associated computer-readable media provide the PC 500 with non-volatile storage of computer-readable instructions, data structures, program modules, and other data. Other types of computer-readable media that can store PC-accessible data, such as magnetic tape cartridges, flash memory cards, digital video discs, CDs, DVDs, RAM, ROM, etc., may also be used in the exemplary operating environment.

[0136] Numerous program modules can be stored in storage device 530, which includes the operating system, one or more applications, other program modules, and program data. Decoding logic can be stored in storage device 530 or on memory 504. Users can input commands and information into PC 500 through one or more input devices 540 (such as a keyboard) and pointing devices (such as a mouse). Other input devices may include digital cameras, microphones, joysticks, gamepads, satellite dishes, scanners, etc. These and other input devices are typically connected to one or more processing units 502 via a serial port interface coupled to system bus 506, but may also be connected via other interfaces such as parallel ports, game ports, or Universal Serial Bus (USB). Monitor 546 or other types of display devices are also connected to system bus 506 via an interface such as a video adapter. Other peripheral output devices 545 may be included, such as speakers and printers (not shown).

[0137] PC 500 can operate in a networked environment using a logical connection to one or more remote computers (such as remote computer 560). In some examples, this includes one or more network or communication connections 550. Remote computer 560 can be another PC, server, router, network PC, or peer device, or other common network node, and typically includes many or all of the elements described above relative to PC 500, although... Figure 5Only the memory storage device 562 is shown in the diagram. The personal computer 500 and / or remote computer 560 can be connected to a logical local area network (LAN) and a wide area network (WAN). This type of network environment is common in offices, enterprise-wide computer networks, intranets, and the Internet.

[0138] When used in a LAN network environment, PC 500 connects to the LAN via a network interface. When used in a WAN network environment, PC 500 typically includes a modem or other devices for establishing communication over the WAN, such as the Internet. In a networked environment, program modules, or portions thereof, depicted on PC 500 may be stored on remote storage devices or other locations on the LAN or WAN. The network connections shown are exemplary, and other means of establishing communication links between computers may be used.

[0139] The example quantum devices disclosed herein include a collimator measurement circuit that implements error-correcting codes using multiple Majorana qubit islands. This collimator measurement circuit is adapted to perform collimator measurements by executing a sequence of measurement-only operations, wherein each measurement-only operation involves up to two Majorana qubit islands.

[0140] In any of the aforementioned exemplary quantum devices, each Majorana qubit island in the Majorana qubit island is either a Majorana quad or a Majorana hexapod.

[0141] In another exemplary quantum device of any of the aforementioned quantum devices, multiple Majorana qubit islands are arranged in a regular array.

[0142] In yet another exemplary quantum device of any of the aforementioned quantum devices, the error-correcting code is a surface code.

[0143] In yet another exemplary quantum device of any of the aforementioned quantum devices, the Majorana qubit island comprises a quadruple representing data qubits and a six-sub-unit representing auxiliary qubits.

[0144] In yet another exemplary quantum device of any of the aforementioned devices, the calibrator measurement comprises a plurality of stabilizer measurements, each of which is achieved by a sequence of joint fermion parity measurements.

[0145] In yet another exemplary quantum device of any of the aforementioned quantum devices, each measurement in the sequence of joint fermion parity checks measures two or four Majorana zero modes.

[0146] In another exemplary quantum device of any of the aforementioned devices, the sequence of joint fermion parity measurements is optimized relative to the measurement resource cost function.

[0147] In yet another exemplary quantum device of any of the aforementioned devices, each stabilizer measurement is influenced by a sequence of eleven joint fermion parity measurements. 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.

[0148] The example method disclosed herein provides a method for implementing a corrector measurement for an error-correcting code by performing a sequence of measurement-only operations on multiple Majorana qubit islands. Each measurement-only operation involves up to two Majorana qubit islands.

[0149] In any of the exemplary methods described above, each Majorana qubit island in the Majorana qubit island is either a Majorana quad or a Majorana hexapod.

[0150] In yet another exemplary method of any of the foregoing methods, multiple Majorana qubit islands are arranged in a regular array.

[0151] In yet another exemplary method of any of the foregoing methods, the error-correcting code is a surface code.

[0152] In yet another exemplary method of any of the foregoing methods, the Majorana qubit island comprises a quadruple representing data qubits and a six-sub-unit representing auxiliary qubits.

[0153] In yet another exemplary method of any of the foregoing methods, implementing the calibrator measurement further includes performing a stabilizing submeasurement for each of the plurality of squares in the surface coding, the stabilizing submeasurement being implemented by a sequence of joint fermion parity measurements.

[0154] In another exemplary approach to any of the foregoing methods, each measurement in the sequence of joint fermion parity checks measures two or four Majorana zero modes.

[0155] In yet another exemplary method of any of the foregoing methods, the sequence of joint fermion parity measurements is optimized relative to the measurement resource cost function.

[0156] In another exemplary method of any of the foregoing methods, the stabilizer measurement is implemented by a measurement sequence consisting of eleven joint fermion parity checks. Seven of the eleven fermion parity checks relate to two Majorana zero modes, and four of the eleven fermion parity checks relate to four Majorana zero modes.

[0157] The exemplary system disclosed herein includes an apparatus for implementing a corrector measurement for an error-correcting code via measurement-only operations by performing a sequence of measurement-only operations on a plurality of Majorana qubit islands. Each measurement-only operation involves up to two Majorana qubit islands.

[0158] Another exemplary quantum device disclosed herein includes a calibrator measurement circuit that performs calibrator measurement of error-correcting codes by performing a sequence of measurement-only operations on six and four sub-sub ...

[0159] In another exemplary quantum device of any of the aforementioned devices, each measurement in the sequence of measurement operations involves at most two islands in an array of Majorana qubit islands.

[0160] The foregoing specification, examples, and data, along with Appendix A, provide a complete description of the structure and use of exemplary implementations. Since many implementations can be made without departing from the spirit and scope of the claimed invention, the appended claims define the invention. Furthermore, structural features of different examples can be combined in yet another implementation without departing from the cited claims.

Claims

1. A quantum device, comprising: A calibrator measurement circuit that uses multiple Majorana qubit islands to implement error correction codes, the calibrator measurement circuit being adapted to perform calibrator measurements by performing measurement-only operations, each of the multiple calibrator measurements being implemented by a sequence of joint fermion parity checks, each of the joint fermion parity checks involving up to two of the Majorana qubit islands.

2. The quantum device according to claim 1, wherein each of the Majorana qubit islands is a Majorana quad or a Majorana hexa-sub.

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

4. The quantum device according to claim 1, wherein the error correction code is a surface code.

5. The quantum device of claim 1, wherein the Majorana qubit island comprises a quadruple representing data qubits and a hexaple representing auxiliary qubits.

6. The quantum device of claim 1, wherein each measurement in the sequence of joint fermion parity checks measures two or four Majorana zero modes.

7. The quantum device of claim 1, wherein the sequence of joint fermion parity checks is optimized relative to the measurement resource cost function.

8. The quantum device of claim 1, wherein each of the stabilizer measurements is implemented by a sequence of eleven joint fermion parity checks, wherein seven of the eleven joint fermion parity checks relate to two Majorana zero modes, and four of the eleven joint fermion parity checks relate to four Majorana zero modes.

9. A method for measuring a calibrator, comprising: The corrector measurement for the error-correcting code is achieved by performing a sequence of stabilizer measurements, each of which is implemented by a sequence of joint fermion parity checks, the sequence of which is implemented by measurement-only operations on a plurality of Majorana qubit islands, each of which involves at most two Majorana qubit islands.

10. The method of claim 9, wherein each of the Majorana qubit islands is a Majorana quad or a Majorana hexapod.

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

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

13. The method of claim 9, wherein the Majorana qubit island comprises a quadruple representing a data qubit and a hexaple representing an auxiliary qubit.

14. The method of claim 9, wherein implementing the calibrator measurement further comprises: Perform a stable sub-measurement for each of the multiple squares in the surface coding.

15. The method of claim 14, wherein each measurement in the sequence of joint fermion parity checks measures two or four Majorana zero modes.

16. The method of claim 14, wherein the sequence of joint fermion parity checks is optimized relative to a measurement resource cost function.

17. The method of claim 14, wherein the stabilizer measurement is performed by a measurement sequence comprising eleven joint fermion parity checks, wherein seven of the eleven joint fermion parity check measurements relate to two Majorana zero modes, and four of the eleven joint fermion parity check measurements relate to four Majorana zero modes.

18. A quantum device, comprising: A calibrator measurement circuit that performs calibrator measurement of an error-correcting code by performing a sequence of measurement-only operations on six and four sub-sub ...

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

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