Handling component failures in surface code circuits

Abstract

Methods, systems, and apparatus for performing a surface code cycle. In one aspect, a method includes, for a quantum computer comprising qubits arranged on a grid, wherein pairs of neighboring qubits are coupled through respective qubit couplers: receiving indications that one or more qubit couplers included in the quantum computer are defective qubit couplers; compiling a quantum circuit using the indications, comprising: for each sub-grid of a predetermined partition of the grid into multiple sub-grids: assigning, constrained on the defective qubit couplers, a first set of entangling operations to squares in the sub-grid that comprise a respective defective qubit coupler; and assigning a second set of entangling operations to remaining squares in the sub-grid according to a pre-determined rule that preserves the distance of the surface code; and performing the surface code cycle, comprising applying the quantum circuit to the plurality of qubits.

Description

PCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1HANDLING COMPONENT FAILURES IN SURFACE CODE CIRCUITSBACKGROUND

[0001] This specification relates to quantum computing.

[0002] Quantum computing provides a means to solve certain problems that cannot be solved in a reasonable period of time using conventional classical computers. These problems include factoring very large numbers into their primes and searching large, unstructured data sets. A number of physical systems are being explored for their use in quantum computing, including ions, spins in semiconductors, and superconducting circuits. However, none of these systems perform sufficiently well to serve directly as computational qubits. For example, single two-state physical systems, which can be used as physical qubits, are not useful in computations because they cannot reliably encode and retain information for long enough periods of time.

[0003] Therefore, scalable quantum computers require quantum error correction. Classical error correction employs redundancy. For example, in the repetition code, information is copied and stored multiple times. If the copies are later found to disagree, it can be determined that an error has occurred, and a majority vote can be taken to recover the information. Copying quantum information is not possible due to the no-cloning theorem. Therefore, quantum error correction codes spread the logical information of one qubit onto an entangled state of multiple physical qubits. The multiple physical qubits are collectively referred to as a logical qubit.

[0004] Surface codes are a family of quantum error correcting codes that are defined on a two-dimensional gnd of qubits. In the surface code, physical qubits are entangled using a sequence of qubit entangling operations, e.g., CNOT gates, with subsequent measurements of the entangled states providing a means for error correction and error detection.SUMMARY

[0005] This specification describes technologies for handling component failures, e.g., qubit or coupler dropouts, in surface code circuits.

[0006] One innovative aspect of the subject matter described in this specification can be implemented in a method for performing a surface code cycle, the method comprising: for a quantum computer comprising a plurality of qubits arranged on a grid, wherein pairs of neighboring qubits included in the plurality of qubits are coupled through respective qubit couplers: receiving indications that one or more qubit couplers included in the quantumPCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 computer are defective qubit couplers; compiling a quantum circuit using the indications, comprising: for each sub-grid of a predetermined partition of the grid into multiple sub-grids: assigning, constrained on the defective qubit couplers, a first set of entangling operations to squares in the sub-grid that comprise a respective defective qubit coupler; and assigning a second set of entangling operations to remaining squares in the sub-grid according to a predetermined rule that preserves the distance of the surface code; and performing the surface code cycle, comprising applying the quantum circuit to the plurality of qubits.

[0007] Other implementations of these aspects include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more classical and quantum computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions.

[0008] The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations the multiple subgrids comprise: a first and a third sub-grid comprising every other column in the grid; and a second and a fourth sub-grid comprising remaining columns in the grid.

[0009] In some implementations the multiple sub-grids comprise at least four sub-grids.

[0010] In some implementations assigning the first set of entangling operations to squares in the sub-grid that comprise a respective defective qubit coupler comprises, for each square that comprises a defective qubit coupler, assigning an entangling operation to one or more edges of the square that represent a non-defective qubit coupler.

[0011] In some implementations assigning the second set of entangling operations to remaining squares in the sub-grid according to a pre-determined rule that preserves the surface code distance comprises assigning the second set of entangling operations to remaining squares in the sub-grid according to a three coupler surface code or a four coupler surface code.

[0012] In some implementations assigning the second set of entangling operations to remaining squares in the sub-grid comprises assigning, unconstrained on the defective qubit couplers, assigning the second set of entangling operations to remaining squares in the subgrid.PCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1

[0013] In some implementations one or more of the defective qubit couplers comprise qubit couplers that are defective due to hard failures.

[0014] In some implementations one or more of the defective qubit couplers comprise qubit couplers that are defective due to soft failures and wherein the method further comprises: monitoring, during a quantum computation, the quantum computer to obtain error rates of respective two-qubit gates included in the quantum computation, wherein each two-qubit gate is implemented using a respective qubit coupler; determining that error rates of one or more of the two qubit gates exceed a predetermined acceptable threshold; and labelling qubit couplers used to implement the one or more two qubit gates as defective.

[0015] In some implementations the entangling operations comprise CNOT operations.

[0016] In some implementations assigning the first set of entangling operations or the second set of entangling operations to squares in the sub-grid further comprises assigning measurement operations to qubits included in the square, wherein the measurement operations are configured to measure surface code stabilizers.

[0017] Another innovative aspect of the subject matter described in this specification can be implemented in a method for performing a surface code cycle, the method comprising, for a quantum computer comprising a plurality of qubits arranged on a grid, wherein pairs of neighboring qubits included in the plurality of qubits are coupled through respective qubit couplers: receiving indications that one or more qubit couplers included in the quantum computer are defective qubit couplers; obtaining data representing a quantum circuit that implements the surface code cycle using the defective qubit couplers; generating data that represents an updated quantum circuit that implements the surface code cycle without using the defective qubit couplers, comprising: removing, from the quantum circuit, entangling operations that require the defective qubit couplers; and for each sub-grid of a predetermined partition of the grid into multiple sub-grids, assigning a set of entangling operations to squares in the sub-grid that comprise a respective defective qubit coupler according to a predetermined rule that preserves the surface code distance; and performing the surface code cycle, comprising applying the updated quantum circuit to the plurality of qubits.

[0018] Other implementations of these aspects include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more classical and quantum computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programsPCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions.

[0019] The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations the quantum circuit that implements the surface code cycle comprises a quantum circuit that implements a three coupler surface code.

[0020] In some implementations the multiple sub-grids comprise: a first and a third sub-grid comprising every other column in the grid; and a second and a fourth sub-grid comprising remaining columns in the grid.

[0021] In some implementations the multiple sub-grids comprise at least four sub-grids.

[0022] In some implementations one or more of the defective qubit couplers comprise qubit couplers that are defective due to hard failures.

[0023] In some implementations one or more of the defective qubit couplers comprise qubit couplers that are defective due to soft failures.

[0024] In some implementations the method further comprises monitoring, during a quantum computation, the quantum computer to obtain error rates of respective two-qubit gates included in the quantum computation, wherein each two-qubit gate is implemented using a respective qubit coupler; determining that error rates of one or more of the two qubit gates exceed a predetermined acceptable threshold; and labelling qubit couplers used to implement the one or more two qubit gates as defective.

[0025] In some implementations the entangling operations comprise CNOT operations.

[0026] Another innovative aspect of the subject matter described in this specification can be implemented in a method for performing a surface code cycle using a quantum computer comprising a plurality of qubits arranged on a square grid, wherein one or more qubits in the plurality of qubits are defective and define one or more holes in the square grid, the performing comprising: measuring a plurality of XXXX and ZZZZ stabilizers in the square grid, wherein the plurality of XXXX and ZZZZ stabilizers comprise compound stabilizers around the one or more holes in the square grid, the measuring comprising: measuring a first set of columns of alternating XXXX and ZZZZ stabilizers in the square grid; measuring a second set of columns of alternating XXXX and ZZZZ stabilizers in the square grid, wherein each column in the second set of columns is adjacent to a respective column in the first set of columns and comprises XXXX and ZZZZ stabilizers with a same orientation as the XXXX andPCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1ZZZZ stabilizers in the respective column in the first set of columns; measuring a third set of columns of alternating XXXX and ZZZZ stabilizers in the square grid, wherein each column in the third set of columns overlaps a respective column in the first set of columns and comprises XXXX and ZZZZ stabilizers that vertically mirror the XXXX and ZZZZ stabilizers in the respective column in the first set of columns; measuring a fourth set of columns of alternating XXXX and ZZZZ stabilizers in the square grid, wherein each column in the fourth set of columns overlaps a respective column in the second set of columns and comprises XXXX and ZZZZ stabilizers that vertically mirror the XXXX and ZZZZ stabilizers in the respective column in the second set of columns; and providing results of measuring the plurality of XXXX and ZZZZ stabilizers to a classical processor for decoding.

[0027] Other implementations of these aspects include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more classical and quantum computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions.

[0028] The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations at least one hole of the one or more holes comprises one defective qubit.

[0029] In some implementations measuring the first, second, third, and fourth set of columns of alternating XXXX and ZZZZ stabilizers comprise measuring the first, second, third, and fourth set of columns of alternating XXXX and ZZZZ stabilizers in order.

[0030] In some implementations measuring the first, second, third, and fourth set of columns of alternating XXXX and ZZZZ stabilizers comprise measuring the second, first, second, third, fourth, and third sets of columns of alternating XXXX and ZZZZ stabilizers in order.

[0031] In some implementations measuring the first, second, third, and fourth set of columns of alternating XXXX and ZZZZ stabilizers comprise measuring the first, second, first, second, third, fourth, third, and fourth sets of columns of alternating XXXX and ZZZZ stabilizers in order.PCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1

[0032] In some implementations at least one hole of the one or more holes comprises two or more defective qubits.

[0033] In some implementations the method further comprises artificially increasing the size of one or more of the holes in the square grid, comprising treating one or more non-defective qubits as defective.

[0034] In some implementations artificially increasing the size of a hole in the square grid comprises generating a square hole by treating one or more non-defective qubits that neighbor the hole as defective.

[0035] In some implementations the compound stabilizers comprise fragments of X or Z stabilizers.

[0036] The subject matter described in this specification can be implemented in particular ways so as to realize one or more of the following advantages.

[0037] The presently described techniques reduce the negative impact of qubit coupler dropouts in a quantum computing system. In particular, a system implementing the presently described techniques can construct quantum circuits that implement the surface code in the presence of qubit coupler dropouts without removing qubits from the code and affecting the code distance.

[0038] In addition, the presently described techniques reduce the negative impact of qubit dropouts (data qubits or measurement qubits) in a quantum computing system. In particular, a system implementing the presently described techniques can construct quantum circuits that implement the surface code in the presence of qubit dropouts with less lost detectors per round of error correction compared to conventional techniques, e.g., those that delete data qubits around a dropped out measurement qubit or delete operations that involve a dropped out data qubit. For example, the presently described techniques lose 1.5 detectors per round instead of 2 or 7 as in conventional techniques.

[0039] In addition, the presently described techniques provides the flexibility to build surface code circuits that are anisotropic and aperiodic, which enable adaptions of the surface code implementation for var ing noise levels. For example, an aperiodic surface code could be used to implement a version of random compiling to increase robustness against some forms of structured noise.

[0040] The details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter w ll become apparent from the description, the drawings, and the claims.PCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1BRIEF DESCRIPTION OF THE DRAWINGS

[0041] FIG. 1 is a block diagram of an example quantum computing system for implementing the coupler surface code.

[0042] FIG. 2 is a block diagram that shows the couplers and operations used to measure multiple columns of alternating XXXX and ZZZZ stabilizers as part of a three coupler code surface cycle.

[0043] FIG. 3 is a block diagram that shows example couplers and operations used to implement the three coupler surface code cycle in four rounds.

[0044] FIG. 4 is a block diagram that shows example shapes and corresponding circuit operations.

[0045] FIG. 5 is a flowchart of an example process for performing a surface code cycle using a constructive coupler dropout strategy.

[0046] FIG. 6A illustrates example couplers included in a quantum circuit compiled using a constructive coupler dropout strategy.

[0047] FIG. 6B shows an implementation of a constructive coupler dropout strategy'.

[0048] FIG. 7A is a flowchart of an example process for performing a surface code cycle using a coupler dropout removal strategy.

[0049] FIG. 7B is a diagram that shows example couplers included in a quantum circuit compiled using a coupler dropout removal strategy.

[0050] FIG. 8 shows two plots that compare performance of known surface code circuits to the presently described circuits for coupler dropouts.

[0051] FIG. 9 is a block diagram that shows other example couplers and operations used to implement the three coupler surface code cycle in four rounds.

[0052] FIG. 10 shows four different measurement schedules for handling a hole that is caused by a single qubit dropout.

[0053] FIG. 11 is a block diagram that shows example holes caused by multiple qubit dropouts.

[0054] FIG. 12 is a block diagram that shows an artificially grown hole and corresponding measurement schedules.

[0055] FIG. 13 is a flowchart of an example process for performing a surface code cycle on an array with one or more qubit dropouts.

[0056] Like reference numbers and designations in the various drawings indicate like elements.PCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1DETAILED DESCRIPTION

[0057] This specification describes circuit schedules for implementing surface code circuits that handle component failures such as qubit dropouts or qubit coupler dropouts. For example, qubit coupler dropouts can be handled without removing qubits from the code. Quantum circuits are compiled using concepts from a three coupler surface code, where shapes of active couplers and entangling operations are assigned to squares of a grid of qubits based on whether the square includes a coupler dropout or not. Qubit dropouts can be handled using three coupler surface code concepts whereby stabilizers are fragmented midcycle instead of at the end of the cycle (the mid-cycle approach measures all stabilizers over multiple rounds without using extra qubits, whereas an end-cycle approach uses additional qubits and attempts to measure all stabilizers in one round).

[0058] FIG. 1 is a block diagram of an example quantum computing system for implementing the surface code. The example system 100 is an example of a system implemented as part of a quantum computing device in which the systems, components and techniques described in this specification can be implemented.

[0059] The system 100 includes multiple qubits 102 in communication with control electronics 104. The qubits 102 are physical qubits, e.g., physical devices that behave as a two-state quantum system. Each qubit can be in a respective quantum state that occupies one or more levels. The levels include two computational levels, e.g., levels 0- and 1-, and one or more non-computational levels that are each higher than the computational qubit levels, e.g., levels 2- and 3-. Population of the higher, non-computational qubit levels can introduce errors in algorithmic operations or quantum computations performed using the qubit. For example, the occupation of qubit levels outside the computational subspace can hamper or prevent the implementation of quantum error correction operations.

[0060] In some implementations the qubits 102 can be superconducting qubits or semiconducting qubits. For example, the qubits 102 can include Xmon qubits, flux qubits, phase qubits, or qubits with frequency interactions. Generally, the qubits 102 are physical devices that are configured to meet basic requirements for quantum computation. For example, the qubits 102 include physical devices that can be initialized, can perform singlequbit rotations, can participate in two-qubit entangling operations, e.g., controlled-NOT (CNOT) gates, can perform a topological version of the Hadamard transformation, e.g., by exchanging their quantum states in a SWAP operation, and can be measured.PCT / US25 / 15675 13 February 2025 (13.O2.2O25)Attorney Docket No. 56113-0538WO1

[0061] The qubits 102 can be arranged in an array. For example, as shown in FIG. 1, in some implementations the qubits 102 can be arranged as a two dimensional array, such as a square grid 110. The example two dimensional grid 110 depicted in FIG. 1 includes 11 x 7 = 77 qubits, however in some implementations the system 100 may include a smaller or a larger number of qubits. In other implementations the qubits 102 can be arranged on another shaped grid, e.g., a hexagonal grid.

[0062] In some implementations one or more of the qubits can be defective. For example, in some cases one or more of the qubits may be broken and unusable after cooldown.Alternatively, or in addition, one or more of the qubits can become broken or unusable over time. Defective qubits are referred to herein as qubit dropouts 112. The system is configured to compile quantum circuits 114 that handles such qubit dropouts, as described in more detail below.

[0063] The qubits 102 can interact with each other through multiple qubit couplers. The qubit couplers can define nearest neighbor interactions between qubits, e.g., such that in a square grid each qubit interacts with at most four neighboring qubits or in a hexagonal grid each qubit interacts with at most three neighboring qubits. The couplers can, in principle, be any type of coupler, e.g., a capacitive or inductive coupler. In some implementations the strengths of the couplers can be controllable, e.g., frequency controllable. In other implementations the couplers can be couplers with a fixed coupling strength.

[0064] In some implementations one or more of the qubit couplers can be defective due to hard or soft failures. For example, in some cases one or more of the qubit couplers may be broken and unusable after cooldown. Alternatively, or in addition, one or more of the qubit couplers can become broken or unusable over time. As another example, in some cases two- qubit gates implemented using a respective qubit coupler might be erroneous, indicating a potential underlying failure of the qubit coupler. Defective qubit couplers are referred to herein as qubit coupler dropouts 116. The system is configured to compile quantum circuits that handles such qubit coupler dropouts 114, as described in more detail below.

[0065] The control electronics 104 include control devices, e.g., arbitrary waveform generators, that can operate the multiple qubits 102. For example, the control electronics 104 can include control devices that tune operating frequencies of the qubits 102 by applying control signals, e.g., voltage pulses, to the qubits through respective control lines.

[0066] As another example, the control electronics 104 can control individual frequencies of the qubits 102 such that the frequency of one or more of the qubits are adjusted towards or away from a frequency of an excitation pulse generated by an excitation pulse generator onPCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 an excitation driveline. The excitation pulses can include pulses with frequencies that implement quantum operations, e.g., quantum logic gates. The qubits 102 can be coupled to an excitation driveline via respective couplers. In some cases, the couplers can be capacitive couplers, e.g., realized by a micro wave line running adjacent to a qubit capacitor.

[0067] The control electronics 104 can also include control devices that tune frequencies of the couplers that couple the multiple qubits 102.

[0068] The type of control electronics 104 that the system 100 utilizes is dependent on the type of qubits the system uses. As an example, qubits that are realized via atomic, molecular or solid-state quantum systems typically have energy separation of the relevant qubit levels in the microwave or optical domain. The states of such qubits may be manipulated and controlled using external fields, such as microwave or optical fields. In such cases, as an example, mode-locked lasers may serve as control electronics due to their broad-band optical spectra that feature both radio frequency and microwave structure. In another example, the control electronics 104 could include a collection of individual qubit controllers realized by a radio frequency generator as well as one or a collection of global excitation controllers realized by a radio frequency or microwave generator. In both cases, the control electronics 104 can be operated manually or connected to a computer and controlled via suitable software allowing for specifying and automatically running the required qubit operations.

[0069] The system 100 can program the control electronics 104 to implement the surface code. To implement the surface code, each qubit in the multiple qubits 102 has one of two functional types: data qubits, e.g., qubit 106, and measure qubits, e.g., qubit 108. A data qubit, e.g., qubit 106, is a qubit that participates in quantum computations performed by the system 100 and stores quantum information corresponding to the quantum computations. That is, the state of the data qubit encodes logical information for a quantum computation. A measure qubit is a qubit that is used to determine an outcome of a computation performed by the data qubit. For example, during a computation an unknown state of the data qubit can be entangled with the state of the measure qubit using a suitable physical operation, after which the measure qubit can be measured. The measure qubits can include measure-X qubits and measure-Y qubits.

[0070] In conventional implementations of the surface code, each data qubit is directly coupled to multiple measure qubits (and is not directly coupled to any other data qubits) and each measure qubit is directly coupled to multiple data qubits (and is not directly coupled to any other measure qubits). For example, each measurement qubit is coupled to four data qubits (if the measure qubit is in the bulk, if it is at the boundary it is coupled to less data ioPCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 qubits). Each data qubit is coupled to two measure-Z qubits and to two measure-X qubits (if the data qubit is in the bulk, if the data qubit is at the boundary it is coupled to less measure qubits). Therefore, conventional implementations of the surface code typically assume that the quantum hardware includes a square grid of qubits with four couplers per qubit (in the bulk).

[0071] A measure-Z qubit is a qubit that can be used to force its neighboring data qubits a, b, c and d into an eigenstate of the operator product ZaZbZcZdwhere Zarepresents a Pauli-Z operator acting on qubit a. Each measure-Z qubit is therefore described as measuring a ZZZZ stabilizer. In the example array 102 of FIG. 1, ZZZZ stabilizers are represented by darker squares, where data qubits exist at the vertices of the ZZZZ stabilizers and measure-Z qubits exist at the center of each ZZZZ stabilizer. A measure-X qubit is a qubit that can be used to force its neighboring data qubits a, b, c and d into an eigenstate of the operator product XaXbXcXdwhere Xarepresents a Pauli-X operator acting on qubit a. Each measure- X qubit is therefore referred to as measuring a XXXX stabilizer. In the example array 102 of FIG. 1, XXXX stabilizers are represented by lighter squares, where data qubits exist at the vertices of the XXXX stabilizers and measure-X qubits exist at the center of each XXXX stabilizer.

[0072] In conventional implementations of the surface code, the control electronics 104 operate the measure-Z and measure-X qubits by repeatedly applying a quantum circuit to the measure-Z and measure-X qubits and their neighboring data qubits. Each application of a quantum circuit performs one surface code cycle. Each quantum circuit includes a sequence of operations. First, the measure qubit is initialized, e.g., in its ground state. Then, four entangling operations, e.g., CNOT gates or CZ gates, are performed. For a measure-Z qubit, each of the four entangling operations targets the measure qubit and each nearest-neighbor data qubit acts as a control for a respective entangling operation. For a measure-X qubit, each of the four entangling operations targets a respective nearest-neighbor data qubit and the measure-X qubit acts as a control for each of the four entangling operations. In this case, the sequence of operations also includes a Hadamard gate applied to the measure qubit before and after the entangling operations. As described above, conventional implementations of the surface code typically assume that the quantum hardware includes a square grid of qubits with four couplers per qubit (in the bulk), where each of the four couplers is used to perform a respective entangling operation. After the entangling operations are performed, thePCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 measure qubit is measured, e.g., through projective measurement. Following the measurement, a subsequent surface code cycle is performed.

[0073] During execution of the surface code (conventional implementations or the three coupler surface code described below), measurement results can be provided to a classical processor for decoding. The classical processor can implement a decoder that is configured to execute a decoding algorithm to decode the measurement data, i.e., predict which errors may have occurred during a quantum computation performed by the quantum computing system.

[0074] Another implementation of the surface code is the three coupler surface code. The three coupler surface code requires at most three couplers per qubit and can therefore be implemented on square grids, e.g., by turning some nearest neighbor couplers off, or can be implemented on devices with hex grid connectivity (where qubits have 2 or 3 neighbors and no 3-neighbor qubits are adjacent). The three coupler surface code cycle is implemented in two rounds: in the first round a first column of alternating XXXX and ZZZZ stabilizers in the array of qubits is measured. In the second round a second column of alternating XXXX and ZZZZ stabilizers in the array of qubits is measured, where the second column is adjacent to the first column and the XXXX and ZZZZ stabilizer measurements in the second column vertically mirror the XXXX and ZZZZ stabilizer measurements in the first column (that is stabilizer measurements in the first and second column include a respective series of measurements obtained along a path in the qubit array, where when the path of the second column is juxtaposed against the first column, the second column appears to be reflected across a vertical plane). The first and second columns can be repeated throughout the array of qubits in the physical device to measure multiple columns in the array.

[0075] FIG. 2 is a block diagram 200 that shows the couplers and operations used to measure multiple columns of alternating XXXX and ZZZZ stabilizers as part of a three coupler code surface cycle. To measure a first column of alternating XXXX and ZZZZ stabilizers, e.g., column 202, individual quantum circuits for measuring a XXXX stabilizer and a ZZZZ stabilizer are overlapped to produce a corresponding quantum circuit. The quantum circuit includes multiple layers of operations, which can be implemented in a physical device, e.g., a device that includes an array of qubits as described in FIG. 1, according to example circuit schedule 204. In the example shown in FIG. 2, it is assumed that measure-X qubits are located at the top left comers of the XXXX stabilizers and that measure-Z qubits are located atPCT / US25 / 15675 13 February 2025 (13.O2.2O25)Attorney Docket No. 56113-0538WO1 the top right comers of the ZZZZ stabilizers. However, flipping inner CNOT gates (e.g., those performed at steps 2 and 4 described below) can flip the role of the qubits such that the measure-X qubits are located at the bottom left comers of the XXXX stabilizers and the measure-Z qubits are located at the bottom right comers of the ZZZZ stabilizers. In either case, the couplers used to measure the column of alternating XXXX and ZZZZ stabilizers 202 remains the same.

[0076] In a first step of the circuit schedule 204, multiple CNOT gates are performed. A CNOT gate is performed between each measure-X qubit and a data qubit to which the measure-X qubit is coupled (in this non-limiting example, the data qubit to the right of the measure-X qubit in the array of qubits), where the measure-X qubit acts as a control. In parallel, a CNOT gate is performed between each measure-Z qubit and a data qubit to which the measure-Z qubit is coupled (in this non-limiting example, the data qubit to the left of the measure-Z qubit in the array of qubits), where the measure-Z qubit acts are the target. These operations are labelled “1” in the example circuit schedule 204.

[0077] In a second step of the circuit schedule 204, another set of CNOT gates are performed. A CNOT gate is performed between each measure-X qubit and another data qubit to which the measure-X is coupled (in this non-limiting example, the data qubit below the measure-X qubit in the array of qubits), where the measure-X qubit acts as a control. In parallel, a CNOT gate is performed between each measure-Z qubit and another data qubit to which the measure-Z qubit is coupled (in this non-limiting example, the data qubit below the measure-Z qubit in the array of qubits), where the measure-Z qubit acts are the target. These operations are labelled “2” in the example circuit schedule 204.

[0078] In a third step of the circuit schedule 204, measurement operations are performed on each measure-X qubit and each measure-Z qubit. These operations are labelled “3” in the example circuit schedule 204. The fourth step of the circuit schedule 204 is the same as the second step of the circuit schedule 204. The operations included in the fourth step are labelled “4” in the example circuit schedule 204. The fifth step of the circuit schedule 204 is the same as the first step of the circuit schedule 204. The operations included in the fifth step are labelled “5” in the example circuit schedule 204. As shown, the example circuit schedule 204 uses a path of couplers, e.g., including coupler 206, that winds through the column 202.

[0079] The column of alternating XXXX and ZZZZ stabilizers 202 can be repeated throughout the array of qubits in the physical device to measure multiple columns in the array. Since packing the coupler paths together such that stabilizers in every column in thePCT / US25 / 15675 13 February 2025 (13.O2.2O25)Attorney Docket No. 56113-0538WO1 array are measured according to the circuit schedule 204 of FIG. 2 would cause the columns to interfere with each other, the coupler paths can be used to measure stabilizers in every other column of the array, as shown in box 208.

[0080] To measure the stabilizers in the remaining columns of the array without causing interference between columns, additional coupler paths that mirror (along the vertical) the coupler paths shown in box 208 can be added to the array, as shown in box 210. Each additional coupler path corresponds to a circuit schedule 212 that is similar to the circuit schedule 204, where the direction of operations included in the layers of operations are also mirrored, e.g., in the first step “1” of the schedule 212 a CNOT gate is performed between each measure-X qubit in the column and a data qubit to the left of the measure-X qubit instead of the right. Combining the coupler paths (and underlying circuit schedules) shown in boxes 208 and 210 produces the coupler configuration shown in box 214, which corresponds to circuit schedule 216.

[0081] As shown, the three coupler surface code cycle can be implemented in two rounds, where in the first round a first set of columns of alternating XXXX and ZZZZ stabilizers in the array of qubits is measured (e.g., using the couplers in box 208) and in the second round a second set of columns of alternating XXXX and ZZZZ stabilizers in the array of qubits is measured (e.g., using the couplers in box 210). In the present disclosure, the first set of columns are referred to as even columns and the second set of columns are referred to as odd columns, where the surface code circuit alternates between measuring even columns and odd columns. However, the number of rounds can be increased whilst still preserving the surface code, e.g., such that the code distance is maintained and its error detection properties preserved.

[0082] For example, FIG. 3 is a block diagram that shows example couplers and operations used to implement the three coupler surface code cycle in four rounds instead of two. In the example shown in FIG. 3, the first round 302 and the second round 304 are the same as the two rounds described above with reference to FIG. 2. The third round 306 and fourth round 308 are also the same as the two rounds described above wdth reference to FIG. 2. That is, the four rounds shown in FIG. 3 can correspond to two applications of the two rounds described above with reference to FIG. 2. Accordingly, the example shown in FIG. 3 can be used to implement the three coupler surface code. However, in other implementations the third round 306 and fourth round 308 can include different coupler paths and operations, e.g.,PCT / US25 / 15675 13 February 2025 (13.O2.2O25)Attorney Docket No. 56113-0538WO1 coupler paths that wind along rows of the array instead of columns. In these implementations, the multiple rounds implement a four coupler surface code.

[0083] Generally, in cases where the array of qubits is a two-dimensional grid of qubits, each round corresponds to a respective sub-grid (e.g., alternating columns or rows in the two- dimensional grid), where each square in a respective sub-grid includes one or more active couplers that are used to perform entangling operations, e.g., CNOTs, and a qubit that is measured. The active couplers in each square define so-called “shapes”. For example, the squares can include two active couplers that define a “L” shape, three active couplers that define a “U” or “C” shape, or one active coupler that defines an “I”. The number of active couplers in each square is dependent on whether the square is in the bulk or at the boundary (squares at the boundary will have less active couplers) and on the number of rounds (generally, the number of active couplers will reduce as the number of rounds of surface code cycles increases).

[0084] Example shapes and circuit operations represented by the shapes are shown in FIG. 4. Shape 402 includes three active couplers 404a-c and a qubit 406 to be measured. Shape 402 corresponds to circuit 408. Circuit 408 includes the following operations: after the measure qubit 406 is initialized (not shown in FIG. 4), a first CNOT gate 410 is applied to the measure qubit and a first data qubit, where the first data qubit acts as a control. In the present disclosure applying a CNOT gate can include applying a set of quantum operations that has the effect of performing a CNOT operation, e.g., where the set of operations depends on the specific quantum hardware being used to perform the quantum computations. For example, in some implementations the system can apply a CNOT gate indirectly using combinations of CZ gates and Hadamard gates. A second CNOT gate 412 is applied to a second data qubit and a third data qubit, where the third data qubit acts as a control. A third CNOT gate 414 is applied to the measure qubit 406 and the second data qubit, where the second data qubit acts as a control. The measure qubit 406 is measured in the Z basis through application of a measurement operation 416. A fourth CNOT gate 418 is applied to the measure qubit 406 and the second data qubit, where the second data qubit acts as a control. A fifth CNOT gate 420 is applied to the second data qubit and the third data qubit, where the third data qubit acts as a control. A sixth CNOT gate 422 is applied to the measure qubit 406 and the first data qubit, where the first data qubit acts as a control.

[0085] Shape 424 also includes three active couplers and a qubit to be measured. Shape 424 corresponds to circuit 426. Circuit 426 includes the following operations: after the measurePCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 qubit 406 is initialized (not shown in FIG. 4), a first CNOT gate 428 is applied to a first data qubit and a second data qubit, where the second data qubit acts as a control. A second CNOT gate 430 is applied to a third data qubit and the measure qubit, where the measure qubit acts as a control. A third CNOT gate 432 is applied to the second data qubit and the measure qubit, where the measure qubit acts as a control. The measure qubit is measured in the X basis through application of a measurement operation 434. A fourth CNOT gate 436 is applied to the second data qubit and the measure qubit, where the measure qubit acts as a control. A fifth CNOT gate 438 is applied to the third data qubit and the measure qubit, where the measure qubit acts as a control. A sixth CNOT gate 440 is applied to the first data qubit and the second data qubit, where the second qubit acts as a control. The circuits 408 and 426 can overlap, e.g., to form part of a coupler path that winds through a column of the grid.

[0086] Shapes 442 and 444 each include two active couplers and a respective qubit to be measured. Shape 442 corresponds to circuit 446 and shape 444 corresponds to circuit 448. Circuits 446 and 448 are sub-circuits of circuits 408 and 426, respectively. Shapes 450 and 452 each include one active coupler and a qubit to be measured. Shape 450 corresponds to circuit 454 and shape 452 corresponds to circuit 456. Circuits 454 and 456 are sub-circuits of circuits 408 and 426, respectively.

[0087] The shapes 402, 424, 443, 444, 450, and 452 shown in FIG. 4 and variations thereof (e.g., rotations or reflections) can be used to compile a surface code quantum circuit that handles qubit coupler dropouts, as described below with reference to FIGS. 5-7.

[0088] FIG. 5 is a flowchart of an example process 500 for performing a surface code cycle using a constructive coupler dropout strategy. For convenience, the process 500 will be described as being performed by components of a quantum computing system. For example, classical control electronics in communication with an array of physical qubits, e.g., the control electronics 104 of FIG. 1, appropriately programmed, can perform example process 500. The array of physical qubits can be arranged as a square grid, where pairs of neighboring qubits are coupled through respective qubit couplers.

[0089] The system receives indications that one or more qubit couplers are defective qubit couplers, e.g., obtains data that labels one or more qubit couplers included in the quantum computing system as defective (step 502). In some implementations one or more of the defective qubit couplers can be qubit couplers that are defective due to hard failures, e.g., qubit couplers that are broken and cannot be used. Alternatively, or in addition, one or morePCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 of the defective qubit couplers can be qubit couplers that are defective due to soft failures, e.g., qubit couplers that implement erroneous two-qubit gates. For example, the system can monitor the quantum computing system, e.g., during a quantum computation, to obtain respective error rates of two-qubit gates included in the quantum computation. The system may identify defective couplers by monitoring performance of one or more qubits. For example, the system can then identify two-qubit gates that have an error rate that exceeds a predetermined acceptable threshold (referred to as erroneous two-qubit gates) and label the qubit couplers that are used to implement the erroneous two-qubit gates as defective. That is, the system treats the qubit couplers that are identified as defective (e.g. those used to implement the erroneous two-qubit gates) as dropped out and unusable.

[0090] The system compiles a quantum circuit using the received indications (step 504). The system partitions the grid of qubits into sub-grids, for example by partitioning the square grid into multiple sub-grids according to a predetermined partiboning rule. In some implementations the number of sub-grids is greater than or equal to four. For example, a first and third sub-grid can include every other column in the grid and a second and fourth subgrid can include remaining columns in the grid, as illustrated in FIG. 3.

[0091] For each sub-grid of the multiple sub-grids, the system assigns a first set of entangling operations, e.g., CNOT gates or iSWAP gates and measurement operations to squares in the sub-grid that include a respective defective qubit coupler (step 504a). The assignment of the first set of entangling operations is constrained on the defective qubit couplers. For example, to assign entangling operations to a square in a sub-grid that includes a respective defective qubit coupler, the system assigns an entangling operadon to one or more edges of the square that represent a non-defective qubit coupler. No entangling operation is assigned to an edge of the square that represents the defective qubit coupler. The number of entangling operations assigned to the edges of the square is dependent on whether the square is at boundary or in the bulk (i.e., has one or three non-defective couplers that can be assigned an entangling operation) and whether the boundary is spiked or not. The assigned entangling operations can form one of the shapes described above with reference to FIG. 4.

[0092] The system then assigns a second set of entangling operations, e.g., CNOT gates or iSWAP gates, and measurement operations to remaining squares in the sub-grid according to a pre-determined rule or code that preserves the distance and error detection capabilities of the surface code (step 504b). The assignment of the second set of entangling operations is not constrained on the defective qubit couplers. For example, the system can assign entangling operations to edges of the remaining squares according to a three coupler surfacePCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 code, e.g., so that the remaining squares correspond to the circuit schedules shown in FIG. 2, or according to a four-coupler surface code (where columns of paths of qubit couplers have a same orientation). Generally, the predetermined rule can be any rule that preserves the surface code, e.g., ensures that, when all rounds have been implemented and the surface code cycle is complete, each measurement qubit has interacted with its neighboring data qubits and been measured.

[0093] The system then applies the compiled quantum circuit to the plurality of qubits to perform the surface code cycle (step 506).

[0094] FIG. 6A illustrates example couplers included in a quantum circuit compiled using a constructive coupler dropout strategy. In the example shown in FIG. 6A, six qubit couplers are defective, e.g., including qubit coupler 602. The square grid is partitioned into four subgrids. The first and third sub-grids include every other column in the grid, e.g., the columns labelled “1” or “3”. The second and fourth sub-grids include remaining columns in the grid, e.g., the columns labelled “2” and "4". Each square of the grid has been assigned active couplers and corresponding entangling operations to measure the required stabilizer operations. As shown, no entangling operations have been assigned to square edges that represent non-defective qubit couplers.

[0095] FIG. 6B shows an example implementation of a constructive coupler dropout strategy. During stage (a), an indication that two qubit couplers are defective is received. During stage (b), the grid is partitioned into four sub-grids. During stage (c), entangling operations are assigned to squares in sub-grid “1” that include a respective defective qubit coupler.Since the squares in sub-grid “1” do not include any defective qubit couplers, no entangling operations are assigned during stage (c). During stage (d), entangling operations are assigned to remaining squares in sub-grid “1” (in this example all squares in sub-grid “1”). In some implementations additional squares can also be assigned entangling operations to increase the density of information, as shown. During stage (e), entangling operations are assigned to squares in sub-grid “2” that include a respective defective qubit coupler. As show n, no entangling operation is assigned to the edge of the square in grid 2 that represents the defective qubit coupler. The remaining stages (f)-(j) are similar to stages (c)-(e). The resulting rounds of active couplers / entangling operations are shown in box 602.PCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1

[0096] FIG. 7A is a flowchart of an example process 700 for performing a surface code cycle using a coupler dropout removal strategy. For convenience, the process 700 will be described as being performed by components of a quantum computing system. For example, classical control electronics in communication with an array of physical qubits, e g., the control electronics 104 of FIG. 1, appropriately programmed, can perform example process 700. The array of physical qubits can be arranged as a square grid, where pairs of neighboring qubits are coupled through respective qubit couplers.

[0097] The system receives an indication that one or more qubit couplers included in the quantum computing system are defective qubit couplers (step 702). Step 702 is similar to step 502 of example process 500 described above, and for brevity details are not repeated.

[0098] The system obtains data representing a quantum circuit that implements the surface code cycle using the defective qubit couplers (step 704). In some implementations the quantum circuit implements a three coupler surface code or a four-coupler surface code.

[0099] The system generates data that represents an updated quantum circuit that implements the surface code cycle without using the defective qubit couplers (step 706). First, the system removes entangling operations that require the defective qubit couplers from the quantum circuit (step 706a). Then, for each sub-grid of a predetermined partition of the grid into multiple sub-grids, the system assigns a set of entangling operations to squares in the sub-grid that include a respective defective qubit coupler according to a pre-determined rule that preserves the surface code distance (step 706b). Step 706b is similar to step 504b of example process 500 described above, and for brevity details are not repeated.

[0100] The system applies the updated quantum circuit to the plurality of qubits to perform the surface code cycle (step 708).

[0101] FIG. 7B is a diagram 750 that shows example couplers included in a quantum circuit compiled using a coupler dropout removal strategy. In the example shown in FIG. 7B, defective qubit couplers are labelled by the dashed lines. The square grid is partitioned into four sub-grids. The first and third sub-grid (top left and bottom left) include every other column in the grid. The second and fourth sub-grids include remaining columns in the grid. In this example, unlike in the example shown in FIG. 6A, the base quantum circuit (corresponding to step 704 of example process 700) is a four-coupler surface code quantum circuit, where each column of paths of couplers has a same orientation instead of alternating in orientation. After operations that require the defective qubit couplers are removed from the quantum circuit, each square of the grid is assigned active couplers and correspondingPCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 entangling operations to measure the required stabilizer operations. As shown, no entangling operations have been assigned to square edges that represent non-defective qubit couplers.

[0102] FIG. 8 shows two plots 800 and 850. Plot 800 shows the performance of four surface code circuits: a standard distance 5 surface code circuit when dropouts are ignored, a standard distance 5 footprint (which becomes a distance 4 circuit) surface code circuit using a known strategy with two coupler dropouts, a distance 5 circuit constructed using the presently described techniques (a “LUCI” circuit) when dropouts are ignored, and a distance 5 LUCI circuit with two coupler dropouts. The x-axis represents error rate, and the y-axis represents logical error rate. Plot 800 shows that a similar performance between standard circuits and the presently described LUCI circuits is obtained when dropouts are ignored, but the presently described LUCI circuits achieve improved performance when dropouts are included. Plot 850 compares the performance of various known distance 7 surface code circuits (dotted lines) to the presently described LUCI circuits (solid lines) for different numbers of coupler dropouts. The x-axis represents error rate, and the y-axis represents logical error rate. As shown, there is far less of a cost of dropping out a qubit coupler using the presently described techniques, as shown by the tighter grouping of solid lines.

[0103] As described above with reference to FIG. 2, the three coupler surface code cycle can be implemented in two rounds, where in the first round a first set of columns of alternating XXXX and ZZZZ stabilizers in the array of qubits is measured (e.g., using the couplers in box 208) and in the second round a second set of columns of alternating XXXX and ZZZZ stabilizers in the array of qubits is measured (e g., using the couplers in box 210). However, the number of rounds can be increased whilst still preserving the surface code.

[0104] For example, FIG. 9 is a block diagram that shows other example couplers and operations used to implement the three coupler surface code cycle in four rounds instead of two. In the example shown in FIG. 9, the first round (labelled “1”) includes measuring a first set of columns of alternating XXXX and TLTL stabilizers in an array of qubits using the couplers in box 208 of FIG. 2.

[0105] The second round (labelled “2”) includes measuring a second set of columns of alternating XXXX and L ' ' LL' stabilizers in the array of qubits using the couplers in box 208 of FIG. 2, where each column in the second set of columns is adjacent to a respective column in the first set of columns. That is, the second set of columns of alternating XXXX and ZZZZ stabilizers have a same shape / orientation as the first set of columns of alternatingPCT / US25 / 15675 13 February 2025 (13.O2.2O25)Attorney Docket No. 56113-0538WO1XXXX and ZZZZ stabilizers but are shifted across by one column to the left or right in the array.

[0106] The third round (labelled “3”) includes measuring a third set of columns of alternating XXXX and ZZZZ stabilizers in the array of qubits using the couplers in box 210 of FIG. 2, where the columns in the third set are the same as the columns in the first set. That is, each column in the third set of columns overlaps a respective column in the first set of columns and includes XXXX and T LL stabilizers that vertically mirror the XXXX and 7 L stabilizers in the respective column in the first set of columns (or equivalently includes a coupler path that vertically mirrors the coupler path in the respective column in the first set of columns).

[0107] The fourth round (“labelled “4”) includes measuring a fourth set of columns of alternating XXXX and ZZZZ stabilizers in the array of qubits using the couplers in box 210 of FIG. 2, where the columns in the fourth set are the same as the columns in the second set. That is, each column in the fourth set of columns overlaps a respective column in the second set of columns and includes XXXX and ZZZZ stabilizers that vertically mirror the XXXX and ZZZZ stabilizers in the respective column in the second set of columns (or equivalently includes a coupler path that vertically mirrors the coupler path in the respective column in the second set of columns). In other words, the fourth set of columns of alternating XXXX and ZZZZ stabilizers have a same shape / orientation as the third set of columns of alternating XXXX and ZZZZ stabilizers but are shifted across by one column to the left or right in the array.

[0108] The example array shown in FIG. 9 includes one qubit dropout 902. The qubit dropout 902 has caused a hole 904 in the array, fragmenting the XXXX and ZZZZ stabilizers around the qubit dropout 902 into fragments 906a, 906b, 908a, 908b. However, by construction, iterating through each of the above described four rounds forms the two compound stabilizers around the hole 904. The first round performs measurements for fragmented stabilizer 906a. The second round performs measurements for fragmented stabilizer 906b which, when combined with the measurements for fragmented stabilizer 906a, forms a first compound stabilizer. The third round performs measurements for fragmented stabilizer 908a. The fourth round performs measurements for fragmented stabilizer 908b which, when combined with the measurements for fragmented stabilizer 908a, forms a second compound stabilizer and closes the hole.

[0109] The order in which the first, second, third, and fourth rounds shown in FIG. 9 are implemented can vary', e.g., so that the rounds are compatible with the surface code circuitPCT / US25 / 15675 13 February 2025 (13.O2.2O25)Attorney Docket No. 56113-0538WO1 schedule. For example, the order in which the rounds are performed varies based on whether a hole starts on an even or odd column in the qubit array (so information about the holes in different column parities comes in at different times) and the type of stabilizers on the leading diagonal of the hole.

[0110] FIG. 10 shows four different measurement schedules that handle respective holes that are caused by single qubit dropouts. The first measurement schedule 1002 handles a first hole in the qubit array that has a first type of stabilizer on the leading diagonal, e.g., ZZZZ stabilizer. The first measurement schedule performs the rounds in the order “0123” (where 0 represents the first round, 1 represents the second round, 2 represents the third round, and 3 represents the fourth round). The second measurement schedule 1004 handles a second hole in the qubit array that has a second type of stabilizer on the leading diagonal, e.g., XXXX stabilizer. The second measurement schedule performs the rounds in the order “2301”. The third measurement schedule 1006 handles a third hole in the qubit array that also has the first type of stabilizer on the leading diagonal. The third measurement schedule performs the rounds in the order “3210”. The fourth measurement schedule 1008 handles a fourth hole in the qubit array that has the second type of stabilizer on the leading diagonal. The fourth measurement schedule performs the rounds in the order “1032”.[OHl] In some implementations one or more of the first, second, third, and fourth rounds can be repeated in each iteration, e.g., to increase the number of detectors obtained per surface code cycle. For example, iterating the pattern 012301230123... produces 0.5 detectors per surface code cycle, whereas iterating the pattern 101232101232... produces 0.666 detectors per surface code cycle and iterating the pattern 0101232301012323... produces 0.75 detectors per surface code cycle.

[0112] In some implementations a hole in the array can be caused by multiple qubit dropouts. FIG. 11 is a block diagram that shows example holes caused by multiple qubit dropouts. For example, holes 1102, 1104, and 1106 are caused by two qubit dropouts (labelled “H”). In these examples the surface code cycle can be implemented using measurement schedules that perform rounds “0123012”, “0123”, and “30123012301.” Hole 1108 is cause by several qubit dropouts. In this example, the surface code cycle can be implemented according to the shown measurement layer indices.PCT / US25 / 15675 13 February 2025 (13.O2.2O25)Attorney Docket No. 56113-0538WO1

[0113] In some implementations it can be beneficial to artificially grow (increase the size of) holes in the array, e.g., by treating one or more non-defective qubits as defective, to simplify the measurement schedule. FIG. 12 is a block diagram that shows an artificially grow n hole and corresponding measurement schedules. The original hole includes two qubit dropouts (labelled “H”). A corresponding measurement schedule for this hole is “30123012301”. This measurement schedule produces 1.4 detectors per cycle.

[0114] The original hole can be artificially grown such that the hole is “squared” by treating two additional qubits as dropouts. A corresponding measurement schedule for this hole is “012301230”. This measurement schedule produces 0.5 detectors per cycle.

[0115] The original hole can be artificially grown such that the hole is “chipped” by treating one additional qubit as a dropout. A corresponding measurement schedule for this hole is “012301230”. This measurement schedule produces 1.0 detectors per cycle.

[0116] FIG. 13 is a flowchart of an example process 1300 for performing a surface code cycle on an array with one or more qubit dropouts. For convenience, the process 1300 will be described as being performed by components of a quantum computing system. For example, classical control electronics in communication with an array of physical qubits, e.g., the control electronics 104 of FIG. 1, appropriately programmed, can perform example process 1300. The array of physical qubits can be arranged as a square gnd, where pairs of neighboring qubits are coupled through respective qubit couplers. One or more qubits in the array of physical qubits are defective and define one or more holes in the square grid.

[0117] The system measures XXXX and ZZZZ stabilizers in the square grid (step 1302). The measured XXXX and ZZZZ stabilizers include compound stabilizers around the one or more holes in the square grid, e.g., stabilizers composed of fragmented X and / or Z stabilizers.

[0118] To measure the XXXX and ZZZZ stabilizers, the system measures a first set of columns of alternating XXXX and ZZZZ stabilizers in the square grid (step 1304). The system then measures a second set of columns of alternating XXXX and ZZZZ stabilizers in the square grid (step 1306). Each column in the second set of columns is adjacent to a respective column in the first set of columns and includes XXXX and ZZZZ stabilizers with a same orientation as the XXXX and ZZZZ stabilizers in the respective column in the first set of columns.

[0119] The system then measures a third set of columns of alternating XXXX and ZZZZ stabilizers in the square grid (step 1308). Each column in the third set of columns overlaps aPCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 respective column in the first set of columns and includes XXXX and ZZZZ stabilizers that vertically mirror the XXXX and ZZZZ stabilizers in the respective column in the first set of columns.

[0120] The system then measures a fourth set of columns of alternating XXXX and ZZZZ stabilizers in the square grid (step 1310). Each column in the fourth set of columns overlaps a respective column in the second set of columns and includes XXXX and ZZZZ stabilizers that vertically mirror the XXXX and ZZZZ stabilizers in the respective column in the second set of columns.

[0121] The system provides results of measuring the plurality of XXXX and ZZZZ stabilizers to a classical processor for decoding (step 1312).

[0122] In some implementations the techniques disclosed herein for handling qubit coupler dropouts and qubit dropouts can be combined, e.g., the techniques described above with reference to FIGS. 9-13 can be combined with the techniques described above with reference to FIGS. 4-8.

[0123] Implementations of the subject matter and operations described in this specification can be implemented in digital electronic circuitry, analog electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-embodied software or firmware, in computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The term “quantum computational systems” may include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0124] Implementations of the subject matter described in this specification can be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, data processing apparatus. The computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively, or in addition, the program instructions can be encoded on an artificially- generated propagated signal that is capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated toPCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus.

[0125] The terms quantum information and quantum data refer to information or data that is carried by, held or stored in quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two- level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states are possible.

[0126] The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0127] A digital computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled orPCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL or Quipper.

[0128] A computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, subprograms, or portions of code. A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data.

[0129] The processes and logic flows described in this specification can be performed by one or more programmable computers, operating with one or more processors, as appropriate, executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers.

[0130] For a system of one or more computers to be “configured to” perform particular operations or actions means that the system has installed on its software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by data processing apparatus, cause the apparatus to perform the operations or actions. For example, a quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions.

[0131] Computers suitable for the execution of a computer program can be based on general or special purpose processors, or any other kind of central processing unit. Generally, a central processing unit will receive instructions and data from a read-only memory, a randomPCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof.

[0132] The elements of a computer include a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital, analog, and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a computer need not have such devices.

[0133] Quantum circuit elements (also referred to as quantum computing circuit elements) include circuit elements for performing quantum processing operations. That is, the quantum circuit elements are configured to make use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data in a non-deterministic manner. Certain quantum circuit elements, such as qubits, can be configured to represent and operate on information in more than one state simultaneously. Examples of superconducting quantum circuit elements include circuit elements such as quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), and superconducting quantum interference devices (SQUIDs) (e.g., RF-SQUID or DC-SQUID), among others.

[0134] In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively carry out instructions of a computer program by performing basic arithmetical, logical, and / or input / output operations on data, in which the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to transmit data to and / or receive data from the quantum circuit elements through electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuitry, rapid single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices and ERSFQ devices, which are an energy-efficient version of RSFQ that does not use bias resistors.

[0135] In certain cases, some or all of the quantum and / or classical circuit elements may be implemented using, e.g., superconducting quantum and / or classical circuit elements.Fabrication of the superconducting circuit elements can entail the deposition of one or more materials, such as superconductors, dielectrics and / or metals. Depending on the selected material, these materials can be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxialPCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 techniques, among other deposition processes. Processes for fabricating circuit elements described herein can entail the removal of one or more materials from a device during fabrication. Depending on the material to be removed, the removal process can include, e.g., wet etching techniques, dry etching techniques, or lift-off processes. The materials forming the circuit elements described herein can be patterned using known lithographic techniques (e.g., photolithography or e-beam lithography).

[0136] During operation of a quantum computational system that uses superconducting quantum circuit elements and / or superconducting classical circuit elements, such as the circuit elements described herein, the superconducting circuit elements are cooled down within a cryostat to temperatures that allow a superconductor material to exhibit superconducting properties. A superconductor (alternatively superconducting) material can be understood as material that exhibits superconducting properties at or below a superconducting critical temperature. Examples of superconducting material include aluminum (superconductive critical temperature of 1.2 kelvin) and niobium (superconducting critical temperature of 9.3 kelvin). Accordingly, superconducting structures, such as superconducting traces and superconducting ground planes, are formed from material that exhibits superconducting properties at or below a superconducting critical temperature.

[0137] In certain implementations, control signals for the quantum circuit elements (e.g., qubits and qubit couplers) may be provided using classical circuit elements that are electrically and / or electromagnetically coupled to the quantum circuit elements. The control signals may be provided in digital and / or analog form.

[0138] Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magnetooptical disks; CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence.

[0139] Control of the various systems described in this specification, or portions of them, can be implemented in a computer program product that includes instructions that are stored on one or more non-transitory machine-readable storage media, and that are executable on one or more processing devices. The systems described in this specification, or portions of them,PCT / US25 / 15675 13 February 2025 (13.02.2025)Attorney Docket No. 56113-0538WO1 can each be implemented as an apparatus, method, or system that may include one or more processing devices and memory to store executable instructions to perform the operations described in this specification.

[0140] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0141] Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations described above should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0142] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

Claims

Attorney Docket No. 56113-0538WO1CLAIMSWhat is claimed is:

1. A method for performing a surface code cycle, the method comprising: for a quantum computer comprising a plurality of qubits arranged on a grid, wherein pairs of neighboring qubits included in the plurality of qubits are coupled through respective qubit couplers: receiving indications that one or more qubit couplers included in the quantum computer are defective qubit couplers; compiling a quantum circuit using the indications, comprising: for each sub-grid of a predetermined partition of the grid into multiple sub-grids: assigning, constrained on the defective qubit couplers, a first set of entangling operations to squares in the sub-grid that comprise a respective defective qubit coupler; and assigning a second set of entangling operations to remaining squares in the sub-grid according to a pre-determined rule that preserves the distance of the surface code; and performing the surface code cycle, comprising applying the quantum circuit to the plurality of qubits.

2. The method of claim 1, wherein the multiple sub-grids comprise: a first and a third sub-grid comprising every other column in the grid; and a second and a fourth sub-grid comprising remaining columns in the grid.

3. The method of claim 1 or claim 2, wherein the multiple sub-grids comprise at least four sub-grids.

4. The method of any one of the preceding claims, wherein assigning the first set of entangling operations to squares in the sub-grid that comprise a respective defective qubit coupler comprises, for each square that comprises a defective qubit coupler, assigning an entangling operation to one or more edges of the square that represent a non-defective qubit coupler.Attorney Docket No. 56113-0538WO15. The method of any one of the preceding claims, wherein assigning the second set of entangling operations to remaining squares in the sub-grid according to a pre-determined rule that preserves the surface code distance comprises assigning the second set of entangling operations to remaining squares in the sub-grid according to a three coupler surface code or a four coupler surface code.

6. The method of any one of the preceding claims, wherein assigning the second set of entangling operations to remaining squares in the sub-grid comprises assigning, unconstrained on the defective qubit couplers, assigning the second set of entangling operations to remaining squares in the sub-grid.

7. The method of any one of the preceding claims, wherein one or more of the defective qubit couplers comprise qubit couplers that are defective due to hard failures.

8. The method of any one of the preceding claims, wherein one or more of the defective qubit couplers comprise qubit couplers that are defective due to soft failures and wherein the method further comprises: monitoring, during a quantum computation, the quantum computer to obtain error rates of respective two-qubit gates included in the quantum computation, wherein each two- qubit gate is implemented using a respective qubit coupler; determining that error rates of one or more of the tw o qubit gates exceed a predetermined acceptable threshold; and labelling qubit couplers used to implement the one or more tw o qubit gates as defective.

9. The method of any one of the preceding claims, wherein the entangling operations comprise CNOT operations.

10. The method of any one of the preceding claims, wherein assigning the first set of entangling operations or the second set of entangling operations to squares in the sub-grid further comprises assigning measurement operations to qubits included in the square, wherein the measurement operations are configured to measure surface code stabilizers.

11. A quantum computing apparatus comprising:Attorney Docket No. 56113-0538WO1 a plurality' of physical qubits; qubit couplers defining nearest neighbor interactions between the plurality of qubits; and control electronics configured to operate the plurality of qubits and qubit couplers, wherein the control electronics are configured to perform operations for implementing a surface code cycle, the operations comprising: for a quantum computer comprising a plurality- of qubits arranged on a grid, yvherein pairs of neighboring qubits included in the plurality of qubits are coupled through respective qubit couplers: receiving indications that one or more qubit couplers included in the quantum computer are defective qubit couplers; compiling a quantum circuit using the indications, comprising: for each sub-grid of a predetermined partition of the grid into multiple sub-grids: assigning, constrained on the defective qubit couplers, a first set of entangling operations to squares in the sub-grid that comprise a respective defective qubit coupler; and assigning a second set of entangling operations to remaining squares in the sub-grid according to a pre-determined rule that preserves the distance of the surface code; and performing the surface code cycle, comprising applying the quantum circuit to the plurality7of qubits.

12. A method for performing a surface code cycle, the method comprising: for a quantum computer comprising a plurality- of qubits arranged on a grid, yvherein pairs of neighboring qubits included in the plurality of qubits are coupled through respective qubit couplers: receiving indications that one or more qubit couplers included in the quantum computer are defective qubit couplers; obtaining data representing a quantum circuit that implements the surface code cycle using the defective qubit couplers; generating data that represents an updated quantum circuit that implements the surface code cycle without using the defective qubit couplers, comprising:Attorney Docket No. 56113-0538WO1 removing, from the quantum circuit, entangling operations that require the defective qubit couplers; and for each sub-grid of a predetermined partition of the grid into multiple sub-grids, assigning a set of entangling operations to squares in the sub-grid that comprise a respective defective qubit coupler according to a pre-determined rule that preserves the surface code distance; and performing the surface code cycle, comprising applying the updated quantum circuit to the plurality of qubits.

13. The method of claim 12, wherein the quantum circuit that implements the surface code cycle comprises a quantum circuit that implements a three coupler surface code.

14. The method of claim 12 or claim 13, wherein the multiple sub-grids comprise: a first and a third sub-grid comprising every other column in the grid; and a second and a fourth sub-grid comprising remaining columns in the grid.

15. The method of any one of claims 12 to 14, wherein the multiple sub-grids comprise at least four sub-grids.

16. The method of any one of claims 12 to 15. wherein one or more of the defective qubit couplers comprise qubit couplers that are defective due to hard failures.

17. The method of any one of claims 12 to 16, wherein one or more of the defective qubit couplers comprise qubit couplers that are defective due to soft failures.

18. The method of claim 17, wherein the method further comprises: monitoring, during a quantum computation, the quantum computer to obtain error rates of respective two-qubit gates included in the quantum computation, wherein each two- qubit gate is implemented using a respective qubit coupler; determining that error rates of one or more of the two qubit gates exceed a predetermined acceptable threshold; and labelling qubit couplers used to implement the one or more two qubit gates as defective.Attorney Docket No. 56113-0538WO119. The method of any one of claims 12 to 18, wherein the entangling operations comprise CNOT operations.

20. A quantum computing apparatus comprising: a plurality' of physical qubits; qubit couplers defining nearest neighbor interactions between the plurality of qubits; and control electronics configured to operate the plurality of qubits and qubit couplers, wherein the control electronics are configured to perform operations for implementing a surface code cycle, the operations comprising: for a quantum computer comprising a plurality’ of qubits arranged on a grid, wherein pairs of neighboring qubits included in the plurality of qubits are coupled through respective qubit couplers: receiving indications that one or more qubit couplers included in the quantum computer are defective qubit couplers; obtaining data representing a quantum circuit that implements the surface code cycle using the defective qubit couplers; generating data that represents an updated quantum circuit that implements the surface code cycle without using the defective qubit couplers, comprising: removing, from the quantum circuit, entangling operations that require the defective qubit couplers; and for each sub-grid of a predetermined partition of the grid into multiple sub-grids, assigning a set of entangling operations to squares in the sub-grid that comprise a respective defective qubit coupler according to a pre-determined rule that preserves the surface code distance; and performing the surface code cycle, comprising applying the updated quantum circuit to the plurality of qubits.

21. A method for performing a surface code cycle using a quantum computer comprising a plurality of qubits arranged on a square grid, yvherein one or more qubits in the plurality of qubits are defective and define one or more holes in the square grid, the performing comprising:Attorney Docket No. 56113-0538WO1 measuring a plurality of XXXX and ZZZZ stabilizers in the square grid, wherein the plurality of XXXX and ZZZZ stabilizers comprise compound stabilizers around the one or more holes in the square grid, the measuring comprising: measuring a first set of columns of alternating XXXX and ZZZZ stabilizers in the square grid; measuring a second set of columns of alternating XXXX and ZZZZ stabilizers in the square grid, wherein each column in the second set of columns is adjacent to a respective column in the first set of columns and comprises XXXX and ZZZZ stabilizers with a same orientation as the XXXX and ZZZZ stabilizers in the respective column in the first set of columns; measuring a third set of columns of alternating XXXX and ZZZZ stabilizers in the square grid, wherein each column in the third set of columns overlaps a respective column in the first set of columns and comprises XXXX and ZZZZ stabilizers that vertically mirror the XXXX and ZZZZ stabilizers in the respective column in the first set of columns; measuring a fourth set of columns of alternating XXXX and ZZZZ stabilizers in the square grid, wherein each column in the fourth set of columns overlaps a respective column in the second set of columns and comprises XXXX and ZZZZ stabilizers that vertically mirror the XXXX and ZZZZ stabilizers in the respective column in the second set of columns; and providing results of measuring the plurality of XXXX and ZZZZ stabilizers to a classical processor for decoding.

22. The method of claim 21 , wherein at least one hole of the one or more holes comprises one defective qubit.

23. The method of claim 21 or claim 22. wherein measuring the first, second, third, and fourth set of columns of alternating XXXX and ZZZZ stabilizers comprise measuring the first, second, third, and fourth set of columns of alternating XXXX and ZZZZ stabilizers in order.

24. The method of claim 21 or claim 22. wherein measuring the first, second, third, and fourth set of columns of alternating XXXX and ZZZZ stabilizers comprise measuring the second, first, second, third, fourth, and third sets of columns of alternating XXXX and ZZZZAttorney Docket No. 56113-0538WO1 stabilizers in order.

25. The method of claim 21 or claim 22, wherein measuring the first, second, third, and fourth set of columns of alternating XXXX and ZZZZ stabilizers comprise measuring the first, second, first, second, third, fourth, third, and fourth sets of columns of alternating XXXX and ZZZZ stabilizers in order.

26. The method of any one of claims 21 to 25, wherein at least one hole of the one or more holes comprises two or more defective qubits.

27. The method of any one of claims 21 to 26. further comprising artificially increasing the size of one or more of the holes in the square grid, comprising treating one or more nondefective qubits as defective.

28. The method of claim 27, wherein artificially increasing the size of a hole in the square grid comprises generating a square hole by treating one or more non-defective qubits that neighbor the hole as defective.

29. The method of any one of claims 21 to 28, wherein the compound stabilizers comprise fragments of X or Z stabilizers.

30. A quantum computing apparatus comprising: a plurality of physical qubits; qubit couplers defining nearest neighbor interactions between the plurality of qubits; and control electronics configured to operate the plurality of qubits and qubit couplers, wherein the control electronics are configured to perform operations for performing a surface code cycle using a quantum computer comprising a plurality of qubits arranged on a square grid, wherein one or more qubits in the plurality of qubits are defective and define one or more holes in the square grid, the operations comprising: measuring a plurality of XXXX and ZZZZ stabilizers in the square grid, wherein the plurality of XXXX and ZZZZ stabilizers comprise compound stabilizers around the one or more holes in the square grid, the measuring comprising:Attorney Docket No. 56113-0538WO1 measuring a first set of columns of alternating XXXX and ZZZZ stabilizers in the square grid; measuring a second set of columns of alternating XXXX and ZZZZ stabilizers in the square grid, wherein each column in the second set of columns is adjacent to a respective column in the first set of columns and comprises XXXX and ZZZZ stabilizers with a same orientation as the XXXX and ZZZZ stabilizers in the respective column in the first set of columns; measuring a third set of columns of alternating XXXX and ZZZZ stabilizers in the square grid, wherein each column in the third set of columns overlaps a respective column in the first set of columns and comprises XXXX and ZZZZ stabilizers that vertically mirror the XXXX and ZZZZ stabilizers in the respective column in the first set of columns; measuring a fourth set of columns of alternating XXXX and ZZZZ stabilizers in the square grid, wherein each column in the fourth set of columns overlaps a respective column in the second set of columns and comprises XXXX and ZZZZ stabilizers that vertically mirror the XXXX and ZZZZ stabilizers in the respective column in the second set of columns; and providing results of measuring the plurality of XXXX and ZZZZ stabilizers to a classical processor for decoding.

31. A quantum computing apparatus comprising: a plurality' of physical qubits; qubit couplers defining nearest neighbor interactions between the plurality of qubits; and control electronics configured to operate the plurality of qubits and qubit couplers, wherein the control electronics are configured to perform the method of any one of claims 1 to 10 or 12 to 19 or 21 to 29.

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