Quantum stabiliser measurement

Abstract

Methods and systems for performing a quantum error correction code are disclosed A control system performs stabiliser operations on a register of quantum devices to obtain syndrome data. Each stabiliser operation involves a unitary stage and a measurement stage. Multiple independent outcomes are determined for each stabiliser during each round of stabiliser measurement. Measurement stages of stabiliser operations in one set of stabiliser operations set are performed in parallel with unitary stages of stabiliser operations in a different set. A decoding system receives the syndrome data and determines a correction.

Description

[0001] QUANTUM STABILISER MEASUREMENT

[0002] Field of the invention

[0003] The invention relates to methods and systems for performing quantum error correction.

[0004] Background

[0005] Quantum computers have the potential to perform computations that would be intractable on even the most powerful classical computers.

[0006] Instead of representing information using classical bits, quantum computers generally use qubits that can be in a simultaneous superposition of multiple quantum states. Qubits exhibit much higher error rates than the bits used in classical computers, and quantum computers therefore require the use of quantum error correction in order to identify and correct qubit errors. The inherently delicate nature of quantum states means that quantum error correction is likely to be necessary even once quantum computing technology matures.

[0007] Quantum computation speed will be limited by the rate at which error information can be extracted from qubits. Faster extraction of error information will facilitate reduced error rates (because decoders have more information about possible errors during a given period of time) and faster quantum computation (because error correction can be performed more quickly).

[0008] There is a need for improved methods of extracting error information from quantum devices such as qubits.

[0009] Summary of the invention

[0010] According to a first aspect of the invention, there is provided a quantum computing system configured to perform a quantum error correction code involving multiple stabiliser measurement rounds. The quantum computing system comprises a register of quantum devices; a control system coupled to the register of quantum devices; and a decoding system. The control system is configured to perform a plurality of stabiliser operations on the register of quantum devices to obtain syndrome data representative of an error state of the register of quantum devices. Each stabiliser operation is associated with a stabiliser of the quantum error correction code and involves a unitary stage and a measurement stage in which multiple auxiliary quantum devices in the register of quantum devices are measured to obtain multiple independent outcomes for the stabiliser associated with the stabiliser operation during each stabiliser measurement round. The plurality of stabiliser operations comprises a plurality of sets of stabiliser operations. Measurement stages of stabiliser operations in each set are performed in parallel with unitary stages of stabiliser operations in a different set. The decoding system is configured to receive the syndrome data and decode the syndrome data to determine a correction for the error state. A quantum computing system (also referred to herein as a quantum computer) is a computing system that exploits quantum mechanical phenomena (i.e. using quantum devices) to perform computations. The quantum devices may be any quantum devices capable of storing quantum information (i.e. any devices suitable for encoding information using quantum computational states). The quantum devices may be qubits. Alternatively, the quantum devices may be other devices capable of storing quantum information, such as qudits or qutrits. While the description herein will primarily refer to qubits, any reference herein to qubits should be understood to also encompass other types of quantum devices unless explicitly stated otherwise.

[0011] While there are existing methods that measure multiple auxiliary qubits for each stabiliser, the obtained measurements do not provide multiple independent stabiliser outcomes during each stabiliser measurement round. Instead, the obtained measurements are partial measurement values which must be combined to provide a single independent stabiliser outcome. Accordingly, these existing methods only provide a single independent outcome for each stabiliser during each stabiliser measurement round, unlike the present invention which provides multiple independent outcomes for each stabiliser during each stabiliser measurement round.

[0012] By obtaining multiple independent outcomes during each stabiliser measurement round, the present invention is more robust against measurement errors. This reduces the number of rounds of stabiliser measurements required without using unconditional reset of the syndrome qubits between stabiliser measurement rounds, thereby increasing the speed of quantum error correction and reducing logical error rates. Even when unconditional resets can be performed, by performing measurement stages of stabiliser operations in one set in parallel with unitary stages of stabiliser operations in a different set, the present invention reduces the average time required per round of stabiliser measurement, therefore increasing the speed at which quantum error correction can be performed and reducing the amount of time that qubits are idle (during which errors can accumulate).

[0013] By improving runtime decoding performance, the present invention can perform quantum error correction with improved logical fidelity (i.e. the present invention reduces noise compared to existing approaches). Accordingly, the present invention reduces logical error rates when performing quantum error correction on a quantum computer.

[0014] Quantum error correction occurs at a low level in the quantum stack, so the benefits of the present invention occur at the architecture level of the quantum computer (the error correction occurs at the architecture level and is independent of the data being processed / applications being run) and makes the quantum computer run more efficiently and effectively as a computer (due to reduced logical error rates, faster syndrome extraction and reduced runtime compared to alternative approaches). An independent outcome for a stabiliser (also referred to herein as a full outcome, complete outcome, independent stabiliser outcome, full stabiliser outcome or complete stabiliser outcome) is a stabiliser outcome that does not need to be combined with any other values / outcomes to provide a measurement value (e.g. eigenvalue or corresponding value) for a stabiliser. In other words, an independent outcome for a stabiliser is a value that corresponds to an eigenvalue of a stabiliser (here “corresponds to” should be understood to mean that the independent outcome could be equal to the eigenvalue or could be a value assigned to / mapped to / associated with the eigenvalue, e.g. a one-to-one mapping between eigenvalues and independent outcomes).

[0015] The register of quantum devices may comprise a plurality of data quantum devices for encoding logical states, and a plurality of auxiliary quantum devices. The roles of data quantum devices and auxiliary quantum devices are not necessarily fixed for an entire computation and may be swapped between rounds of syndrome measurement, e.g. using SWAP or ISWAP gates or similar.

[0016] Each unitary stage comprises one or more unitary operations acting upon data quantum devices and auxiliary quantum devices associated with a given stabiliser operation. Each unitary stage may comprise a plurality of entangling gates, e.g. a plurality of entangling gates between an auxiliary quantum device and a data quantum device / another auxiliary quantum device. Each measurement stage (which may also be referred to as a projection stage) comprises one or more measurement operations (e.g. projection operations) acting upon auxiliary quantum devices associated with a given stabiliser operation.

[0017] A stabiliser operation (also referred to herein as a stabiliser measurement operation) is an operation (which may comprise both quantum operations and classical processing) for obtaining a measurement value of a stabiliser.

[0018] One skilled in the art will appreciate that a stabiliser measurement round (which may also be referred to as a syndrome measurement round or syndrome extraction round) is one of a sequence of stabiliser operations. Multiple independent outcomes are obtained during a given stabiliser measurement round if they are associated with the same measurement stage for a given stabiliser operation. A single stabiliser measurement round for a given stabiliser involves a single unitary stage followed by a single measurement stage. One skilled in the art will appreciate that stabiliser measurements can be performed in a staggered / offset manner, i.e. the stabiliser measurement rounds of stabilisers associated with stabiliser operations in different sets may not be in-phase with each other.

[0019] A decoding system (also referred to herein as a decoder) is a classical computing system that decodes syndromes and provides one or both of (i) possible error locations (i.e. which qubits may have experienced an error), and (ii) a correction for the qubit error state (e.g. a correction for one or more encoded logical qubits states). It is possible to determine a correction during decoding without determining error locations, and the correction may be one or more bits representing whether a logical error has occurred for each logical operator of each logical qubit encoded (e.g. for each qubit, there may be a one bit representing whether a logical X error has occurred, and another bit representing whether a logical Z error has occurred). The correction can generally be tracked by a classical computer (e.g. by the decoder or a control system) and does not generally need to be applied to the quantum devices. The decoder may be a dedicated hardware device (e.g. implemented using an FPGA or ASIC or similar) or it may be a software component implemented using a CPU.

[0020] A control system (also referred to herein as a quantum control system) is a classical processing device configured to communicate with quantum devices (e.g. by sending and receiving control signals such as RF pulses) to perform operations such as quantum logic gates and state measurement / readout.

[0021] The quantum error correction code may be any error correction code that decodes errors using syndrome data obtained from stabiliser measurements (e.g. any stabiliser code, such as a surface code or a quantum LDPC code; or a more general QEC code such as a subsystem or Floquet code, etc.). Decoding may be performed using any decoding algorithm suitable for QEC codes, e.g. minimum-weight-perfect-matching, clustering algorithms (such as union-find), belief propagation techniques etc.

[0022] The syndrome data (also referred to herein as a syndrome) may be derived from the independent outcomes, or it may be the raw independent outcomes. The syndrome data may be obtained by classical processing of the independent outcomes for each stabiliser. For example, a surface code syndrome is generally obtained by identifying changes in stabiliser values (e.g. using XOR operations between independent outcomes obtained in successive stabiliser measurement rounds for the same stabiliser). One skilled in the art will appreciate that the manner in which the syndrome data is obtained from the independent outcomes will depend upon the specifics of the quantum error correction code being implemented.

[0023] The quantum computing system (e.g. a control system of the quantum computing system) may be further configured to measure a logical state encoded in the quantum devices and apply the correction to the measured logical state (the correction may be applied at the control system, at the decoding system or at some other subsystem of the quantum computing device, such as a device operating at the algorithmic / application layer of the quantum stack). The plurality of sets of stabiliser operations may comprise one or more sets of X-type stabiliser operations and one or more sets of Z-type stabiliser operations. For example, these may be a set of star stabiliser operations and a set of plaquette stabiliser operations of a surface code. The plurality of sets of stabiliser operations may be non-overlapping sets. The plurality of sets of stabiliser operations may comprise additional sets, such as one or more sets of Y-type stabiliser operations.

[0024] The auxiliary quantum devices may also be referred to as syndrome quantum devices (e.g. syndrome qubits). The auxiliary quantum devices are quantum devices that facilitate measurement of the stabilisers. The auxiliary quantum devices may be associated with one or more stabiliser operations. The role of the auxiliary devices and data quantum devices may be changed during a quantum computation, for example using SWAP gates or similar. Each independent outcome may optionally be obtained from (i.e. associated with or derived from) measurement of multiple auxiliary quantum devices. For example, measurement of multiple auxiliary quantum devices may provide a plurality of partial outcomes that can be combined (e.g. using classical processing) to provide an independent outcome.

[0025] Alternatively, each independent outcome may be obtained from (i.e. associated with or derived from) measurement of a single auxiliary quantum device (e.g. the independent outcome may equal the eigenvalue observed when measuring a single quantum device, or some value assigned to / mapped to / associated with that eigenvalue).

[0026] Measuring multiple auxiliary quantum devices to obtain multiple independent outcomes may comprise measuring the multiple auxiliary quantum devices during the same timestep.

[0027] According to a second aspect of the invention, there is provided a (computer-implemented) method of performing a quantum error correction code involving multiple stabiliser measurement rounds. The method is performed at a quantum computing system comprising a register of quantum devices, a control system coupled to the register of quantum devices, and a decoding system. The method comprises performing, by the control system, a plurality of stabiliser operations on the register of quantum devices to obtain syndrome data representative of an error state of the register of quantum devices. The method also comprises receiving, at the decoding system, the syndrome data; and decoding, at the decoding system, the syndrome data to determine a correction for the error state. Each stabiliser operation is associated with a stabiliser of the quantum error correction code and involves a unitary stage and a measurement stage in which multiple auxiliary quantum devices in the register of quantum devices are measured to obtain multiple independent outcomes for the stabiliser associated with the stabiliser operation during each stabiliser measurement round. The plurality of stabiliser operations comprises a plurality of sets of stabiliser operations, wherein measurement stages of stabiliser operations in each set are performed in parallel with unitary stages of stabiliser operations in a different set.

[0028] The second aspect of the invention provides the same benefits as the first aspect of the invention. Any feature described in combination with the first aspect of the invention may also be combined with the second aspect of the invention.

[0029] According to a third aspect of the invention, there is provided a computer-readable medium (such as a non-transitory computer readable medium) comprising instructions which, when executed by a quantum computing system, cause the quantum computing system to carry out the method of the second aspect of the invention.

[0030] Brief description of the drawings

[0031] Examples of the present invention will now be described in detail with reference to the accompanying drawings, in which:

[0032] Fig. 1 is a schematic of a quantum computing system;

[0033] Fig. 2 shows a surface code patch;

[0034] Figs. 3a and 3b show quantum circuits for measuring surface code stabilisers;

[0035] Fig. 4 shows a gate schedule for performing the quantum circuits of Figs. 3a and 3b;

[0036] Fig. 5 shows another quantum circuit for measuring surface code stabilisers;

[0037] Fig. 6 shows a gate schedule for the quantum circuit of Fig. 5;

[0038] Fig. 7 shows example timings for measuring stabilisers;

[0039] Fig. 8 shows another quantum circuit for measuring stabilisers; and

[0040] Fig. 9 is a flowchart of a method for performing a quantum error correction code.

[0041] Detailed description

[0042] Quantum error correction (QEC) algorithms are used to detect and correct errors at the physical qubit level to mitigate against computational errors at the logical qubit level. QEC is expected to be essential for performing useful computations on early quantum computers, and the delicate nature of qubits means that QEC is likely to remain necessary even once quantum computing hardware matures.

[0043] The goal of QEC is to reduce the effect of noise within a quantum computer: building in redundancies to protect fragile quantum systems. This is achieved with QEC codes that encode a number of logical qubits (one or more) into a larger number of physical qubits. If the error rate of these physical qubits is below a certain threshold associated with the QEC code being used, the logical qubits will exhibit a reduced effective error rate compared to the error rate experienced by the physical qubits. Simply put, each logical qubit will outperform the sum of its parts.

[0044] By interacting (non-destructively) with the encoded quantum state via auxiliary qubits (i.e. additional physical qubits that do not themselves encode logical states), it is possible to determine the signature of errors that have affected the logical state; this signature is known as a syndrome. The process of obtaining this syndrome, known as syndrome extraction, provides only partial information. As such, decoding algorithms are employed to determine likely error occurrences and / or corrections (some decoding algorithms, such as some clustering algorithms, can determine a correction without determining likely error occurrences). These algorithms are typically deployed on classical computing hardware with restricted memory and / or processing capabilities.

[0045] The output of a decoder is a probabilistic prediction. Given the syndrome observed, the decoder outputs an error with high associated probability that explains the syndrome, or alternatively a likely correction that will correct the error. A given family of error correction codes may have a variety of decoding algorithms to choose from; selecting the decoder is a balance between accuracy, speed, and compute budget for decoding. A more accurate decoder will be more effective at producing a best guess for errors / corrections, and this will result in improved logical accuracy of the quantum computation. The decoder is therefore a key element in the performance of the QEC protocol (and therefore the quantum computation as a whole).

[0046] A schematic of an exemplary quantum computing system 100 suitable for performing the method of the present disclosure is shown in Fig. 1. The quantum computing system 100 comprises a quantum processing unit (QPU) 106 comprising a register of physical qubits (unless specified otherwise, reference herein to qubits should be understood to refer to physical qubits rather than logical qubits). The qubits may include data qubits used to encode logical qubit states, and auxiliary qubits (or syndrome qubits) used to perform syndrome extraction (e.g. stabiliser measurements) for QEC. While the exemplary quantum computing system 100 uses qubits, one skilled in the art will appreciate that the invention described herein is also applicable to quantum computing systems that use other quantum devices, such as qutrits and qudits having three or more computational states. Accordingly, it should be understood that any reference herein to qubits is applicable to any type of quantum devices that can be used to encode quantum information.

[0047] The QPU 106 is controlled by a control system 104 that has one or more classical processing elements. The control system 104 transmits control signals (e.g. RF pulses) to qubits in the QPU 106 for performing operations on the qubits (including measurement operations), and it receives measurement information from the qubits. The measurement information may be analogue data signals, or analogue readout signals may alternatively be converted into digital signals before being transmitted to the control system 104 (e.g. the qubits or the QPU 106 may be provided with one or more analogue-to-digital converters). The control system 104 may receive high-level instructions from an algorithmic system or similar (not shown) and convert these high-level instructions (such as logic gates) into low- level qubit instructions (e.g. microwave pulses etc.), which may be in analogue format.

[0048] The quantum computing system 100 also comprises a decoding system 102 (also referred to herein as a decoder). The decoding system 102, which will generally be a classical computing system, receives an error syndrome (also referred to herein as a syndrome or syndrome data) obtained from measurements of syndrome qubits (which may also be referred to herein as auxiliary qubits). The error syndrome may comprise raw analogue measurement data, or it may alternatively be pre-processed (e.g. into digital format) by the control system 104. The decoding system 102 may be connected to the control system 104 and receive the error syndrome via the control system 104 as illustrated in Fig. 1 (potentially via one or more additional intermediary systems). In alternative examples the decoding system 102 could be connected directly to the qubits and receive the error syndrome directly from the QPU 106 (e.g. as raw analogue signals or digital measurement values). The decoding system 102 uses a decoding process / algorithm to decode the error syndrome to determine a correction for an error state of the qubits associated with the error syndrome (i.e. a correction for an error state that causes the measured error syndrome).

[0049] While the decoding system 102 and control system 104 are illustrated as being separate components, it should be understood that they could both be subcomponents of a single system. For example, the decoding system 102 and control system 104 could be hardware subcomponents of a single hardware system (e.g. a QEC stack) responsible for control and decoding (in this arrangement, certain functions, such as RF-signal generation and readout, may be implemented by additional unillustrated components separate from the QEC stack, e.g. by a signal generator that receives a digital gate sequence from the QEC stack and generates analogue control pulses). Alternatively, the decoding system 102 and control system 104 could be separate software components executed on a single hardware system (or across a distributed system). The term “classical computing component” will be used herein to refer to any apparatus capable of performing classical computations (such as a CPU, ASIC, FPGA etc.). It should be understood that classical computing components may include the decoding system 102 and / or the control system 104, in addition to any other apparatus / component of the quantum computing system 100 used to perform classical computations / processing while a quantum computation is being performed on the quantum computing system 100.

[0050] One skilled in the art will appreciate that the quantum computing system 100 may also comprise additional intermediary components positioned between the illustrated components, and that the illustrated components may be connected in a different configuration (e.g. the decoding system 102 may be connected directly to the QPU 106 as previously described).

[0051] In use, quantum computations are executed on the qubits controlled by the control system 104. Due to the fragile nature of quantum information states, non-trivial quantum computations will generally require the use of QEC. Myriad QEC schemes have been proposed (Terhal, B. M. Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307-346 (2015) provides a review of some of the most prominent QEC schemes), and the present disclosure will focus primarily on surface codes, which are stabiliser codes that encode logical qubits in registers of physical qubits using topological properties that arise from the measurements used to obtain error syndromes. However, it should be understood that the teachings of the present disclosure can also be applied to other QEC codes.

[0052] Fig. 2 shows a register of qubits that can be used to perform planar code QEC. The register of qubits includes a plurality of data qubits 202 (represented by shaded circles) and auxiliary (or syndrome) qubits 204 (represented by unshaded circles). The data qubits 202 are the physical qubits used to encode logical quantum states in QEC procedures. The auxiliary qubits 204 are physical qubits used to facilitate measurements in QEC procedures. Connections or couplings between qubits (not shown) enable entangling gates to be performed between the connected qubits. Examples of entangling gates include controlled- X gates (also referred to as CNOT gates) and controlled-Z gates (also referred to as CPHASE gates) and ISWAP gates. Unless indicated otherwise, reference to X, Y and Z herein should be understood to refer to Pauli X, Y and Z operators.

[0053] A QEC planar code patch 206 can be implemented using the register of qubits. While it is often preferable that the physical qubits of a planar code are arranged in a regular array as shown in Fig. 2, one skilled in the art will appreciate that this is not essential for implementing planar codes, and the qubits could alternatively have a different physical arrangement while still achieving the desired connectivity. The illustrated planar code patch 206 encodes a single logical qubit and has code distance three (i.e. an undetectable logical qubit error requires errors on at least three data qubits 202), and it is in a configuration commonly referred to as the “rotated” planar code, which requires fewer physical qubits for a given code distance compared to “unrotated” planar codes.

[0054] The planar code patch 206 is formed of X-type stabilisers 210 (unshared squares and triangles) and Z-type stabilisers 208 (hatched squares and triangles). The X-type stabilisers 210 involve joint Pauli-X measurements on the data qubits 202 at the vertices of the X-type stabilisers 210 (four data qubits 202 for the square stabilisers, and only two data qubits 202 for the triangular stabilisers because the third vertex is occupied by an auxiliary qubit 204). Similarly, the Z-type stabilisers 208 involve joint Pauli-Z measurements on the data qubits 202 at the vertices of the Z-type stabilisers 208. Each stabiliser 208, 210 is associated with a single auxiliary qubit 204 (either in the centre of the respective square, or at the third vertex of the respective triangle). All of the stabilisers of the planar code commute with each other, although one skilled in the art will appreciate that other types of code (such as Floquet codes) may measure non-commuting operations when performing QEC. The auxiliary qubits 204 may be used to measure the value of the stabilisers. Figs. 3a and 3b show quantum circuits for measuring the X-type stabilisers 210 and Z-type stabilisers 208 respectively. In both Figs. 3a and 3b, the top qubits (labelled SA and SB respectively) represent the auxiliary qubits 204 associated with the respective stabiliser, and the other qubits (labelled di-de) represent data qubits 202.

[0055] The stabilisers may also be referred to herein as checks. One skilled in the art will appreciate that QEC checks are not always formed of X and / or Z measurements: QEC codes may also use stabilisers / checks with other types of measurements, for example combinations of X, Y and / or Z operations.

[0056] The stabiliser measurement values obtained during the QEC procedure can be used to infer a syndrome that is representative of errors on the data qubits 202. The syndrome is then provided to the decoding system 102, which analyses the syndrome and tracks errors and / or corrections. Often, it suffices to track the errors and / or corrections classically: it is not generally necessary to apply any corrections to the data qubits 202 (the correction can instead be applied by adjusting any subsequent logical state measurements as necessary). The qubit labels in Figs. 3a and 3b correspond to the qubit labels shown in Fig. 4, which shows an order (referred to herein as a schedule) in which the gates of Figs. 3a and 3b can be performed to facilitate simultaneous measurement of the X-type stabilisers 210 and Z-type stabilisers 208. The numerals 1-4 represent timesteps of the two-qubit gates on Figs. 3a and 3b. In the first timestep, a controlled-X gate is performed between auxiliary qubit SA and data qubit d2, and a controlled-Z gate is performed between auxiliary qubit SB and data qubit d4. The schedule then proceeds following the “Z-shaped” and “N-shaped” orderings depicted in Fig. 4, which ensures that the auxiliary qubits SA and SB do not become inadvertently entangled with each other during the stabiliser measurements.

[0057] The schedule shown in Fig. 4 is not unique, and it is possible to use alternative gate schedules for measuring stabilisers (it is also possible to compile to other native gates, which may include operations other than controlled-Pauli gates).

[0058] While the gate schedule shown in Fig. 4 allows for simultaneous measurement of all stabilisers in a planar code, such a schedule also has several disadvantages. Firstly, measurement of the syndrome qubits generally takes much longer than the two-qubit gates (i.e. the unitary part of the measurement circuit). Consequently, the data qubits spend a relatively long proportion of the stabiliser measurement process in an idle state. Secondly, measurement errors can adversely affect the effective code distance of the QEC code related to time-like (measurement induced) logical failures during operations such as lattice surgery and patch movement when using the schedule of Fig. 4 without performing unconditional reset of the syndrome qubits between subsequent stabiliser measurements (also referred to as mid-circuit resets). This phenomenon occurs because accurate stabiliser measurement relies upon the syndrome qubit being in the correct state at the beginning of the stabiliser measurement process: if an incorrect measurement value was observed during a previous round of syndrome measurement then the measurement value expected in the following round in the absence of errors will also be incorrect. In order to restore the effective code distance, it becomes necessary to either double the number of stabiliser measurements (effectively doubling the amount of time required) or to implement mid-circuit resets, which also incurs additional time overhead and presents engineering challenges.

[0059] The present invention overcomes these disadvantages by using multiple syndrome qubits per stabiliser in order to obtain two independent outcomes in parallel for each stabiliser during each round of QEC, thereby restoring the effective code distance with lower time overhead than the existing approaches.

[0060] An example of a quantum circuit 500 for performing stabiliser measurement is shown in Fig. 5. The quantum circuit 500 involves four syndrome qubits S1-S4 and six data qubits di-de, and it utilises multiple syndrome qubits (auxiliary qubits) for each stabiliser operation to provide multiple independent outcomes for each stabiliser during each stabiliser measurement round.

[0061] An independent stabiliser outcome (also referred to herein as a full stabiliser outcome or a complete stabiliser outcome) is a stabiliser outcome that does not need to be combined with any other values / outcomes to provide a measurement value for a stabiliser; this is in contrast to partial stabiliser outcomes which need to be combined to provide an independent stabiliser outcome. In other words, an independent stabiliser outcome is a value that corresponds to an eigenvalue of a stabiliser (e.g. it may be equal to an eigenvalue, or some value uniquely associated with the eigenvalue / eigenstate in a one-to-one mapping). It should be understood that an independent outcome may be obtained from a single qubit measurement or by combining measurement values (i.e. partial outcomes) from multiple qubits.

[0062] The illustrated quantum circuit 500 includes several stages:

[0063] • a first unitary stage 502a comprising a plurality of CNOT and controlled-Z gates acting upon a first pair of syndrome qubits Si , S2 and four data qubits di-d4; • a second unitary stage 502b comprising a plurality of CNOT gates acting upon a second pair of syndrome qubits S3, S4 and four data qubits ds-de;

[0064] • a first measurement stage 504a comprising measurement operations on the second pair of syndrome qubits S3, S4; and

[0065] • a second measurement stage 504b comprising measurement operations on the first pair of syndrome qubits Si , S2.

[0066] The numbers above the first unitary stage 502a and below the second unitary stage 502b correspond to timesteps in the quantum circuit 500.

[0067] In general, the quantum circuit 500 will be part of a series of similar quantum circuits in sequence of stabiliser measurement rounds. Each first measurement stage 504a provides two independent outcomes for a given stabiliser (in this case, an X-type stabiliser ® X ® X ® X acting on data qubits ds-de), and each second measurement stage 504b provides two independent outcomes for a different stabiliser (in this case, a Z-type stabiliser Z ® Z ® Z ® Z acting on data qubits di-d4).

[0068] The unitary stages 502a, 502b act to entangle the syndrome and data qubits such that, at the end of each unitary stage 502a, 502b, an independent outcome is encoded in each associated syndrome qubit (i.e. an independent outcome for the X-type stabiliser is encoded in each syndrome qubit S3, S4 at the end of the second unitary stage 502b, and an independent outcome for the Z-type stabiliser is encoded in each syndrome qubit Si , S2 at the end of the first unitary stage 502a). In the illustrated quantum circuit 500, the first and second unitary stages 502a, 502b essentially each perform two concurrent stabiliser measurement operations similar to those shown in Figs. 3b and 3a respectively with two syndrome qubits that are “swapped” (i.e. their quantum states are swapped using three CNOT gates between the syndrome qubits; they are not physically swapped) during the circuit.

[0069] While the quantum circuit 500 in Fig. 5 utilises CNOT gates to effectively perform SWAP gates between multiple syndrome qubits for surface code stabiliser measurement, one skilled in the art will appreciate that specifics of the stabiliser measurement circuit will depend upon the capabilities of the QPU (e.g. qubit connectivity etc.) and the QEC code being implemented. For example, where device connectivity allows, the SWAP operation could be omitted and each syndrome qubit could directly interact with all associated data qubits (rather than only directly interacting with two data qubits, as shown in Fig. 5).

[0070] It should be understood that the stages associated with the X-type and Z-type stabilisers are staggered such that the illustrated first measurement stage 504a will provide stabiliser measurement values resulting from a second unitary stage 502b in an immediately preceding iteration of the quantum circuit 500. Similarly, the stabiliser measurement values resulting from the illustrated second unitary stage 502b will be measured during the next iteration of the quantum circuit 500. One skilled in the art will appreciate that the first and final iterations of the quantum circuit may therefore be adapted to account for this staggering, e.g. by omitting the first measurement stage 504a during a first iteration and omitting the second unitary stage 502b during a final iteration.

[0071] In general there may be a large number of stabiliser operations during each stabiliser measurement round. The illustrated quantum circuit 500 is suitable for performing unitary stages of all X-type stabilisers of a planar code in parallel with the measurement stages of all Z-type stabilisers and vice-versa, although one skilled in the art will appreciate that alternative quantum circuits could be used for stabiliser measurement and that the illustrated circuit may be adapted depending upon hardware capabilities and details of the QEC code being implemented. In addition, one skilled in the art will appreciate that while the illustrated quantum circuit 500 has two syndrome qubits associated with each stabiliser operation, alternative quantum circuits may be implemented in which additional syndrome qubits are used to provide further independent outcomes for each stabiliser during each stabiliser measurement round.

[0072] Fig. 6 illustrates a schedule for the quantum circuit 500 of Fig. 5 for an X-type stabiliser 210 neighbouring a Z-type stabiliser 208 in a planar code. The labels of the data qubits 202 and auxiliary qubits 204 correspond to those in Fig. 5, and the numbers next to the CNOT and controlled-Z gates correspond to the timesteps in Fig. 5. The schedule can be tiled throughout a surface code such that, during each iteration of the quantum circuit 500, the first unitary stages 502a of all Z-type stabilisers may be performed in parallel with the first measurement stages 504a of all X-type stabilisers, and the second unitary stages 502b of all X-type stabilisers may be performed in parallel with the second measurement stages 504b of all Z-type stabilisers.

[0073] Advantageously, the circuit of Fig. 5 provides two independent outcomes for each stabiliser during each round of stabiliser measurement, which means that the effective code distance can be maintained without requiring unconditional reset or repeated stabiliser operations. The benefits of this approach are shown in Fig. 7, which provides an illustration of the effective time required using different approaches to stabiliser measurement. It should be understood that the timings in Fig. 7 are for illustrative purposes only and are not necessarily to scale.

[0074] The top two rows 702 illustrate the effective time required using a conventional syndrome extraction circuit (such as that described above in relation to Figs. 3a, 3b and 4) without mid-circuit resets. In order to maintain the effective code distance, it is necessary to repeat each stabiliser measurement operation (or equivalently, to double the number of stabiliser measurement rounds), thereby effectively doubling the time required. The middle two rows 704 illustrate the effective time required using a conventional syndrome extraction circuit with mid-circuit resets. While it is not necessary to repeat each stabiliser measurement operation, it is instead necessary to reset the syndrome qubits at the end of each stabiliser measurement round; this means that the mid-circuit approach in the two middle rows 704 is faster than the conventional approach without mid-circuit reset in the top two rows 702, but it still incurs a time penalty.

[0075] The effective time required when using the quantum circuit 500 of Fig. 5 is shown in the bottom two rows 706. While the unitary part of each stabiliser measurement operation requires more timesteps (seven timesteps of unitary gates) compared to the circuits in Figs. 3a and 3b (which require four timesteps of unitary gates), unitary operations are much faster than measurement and reset operations, so the overall time required by the quantum circuit 500 of Fig. 5 is less than that required by the other approaches. In addition, performing measurement stages of one set of stabiliser operations in parallel with unitary stages of another ensures that syndrome qubits in different sets of stabiliser operations do not become entangled (which would require careful scheduling and additional time to avoid when using conventional syndrome measurement techniques).

[0076] In other words, the present invention advantageously utilises additional syndrome qubits to reduce the overall time required for stabiliser measurement by obtaining multiple independent outcomes for each stabiliser by measuring multiple syndrome (auxiliary) qubits for each stabiliser during each stabiliser measurement round and performing the unitary stages in parallel. A synergistic benefit of a reduction in the time required for stabiliser measurement arises from (i) using syndrome qubits to obtain multiple independent stabiliser outcomes per stabiliser measurement round, and (ii) performing measurement stages and unitary stages of different sets of stabiliser operations in parallel; this is because (i) alone would generally double the time required by the unitary stages to avoid inadvertently entangling syndrome qubits from different sets of stabiliser operations, and (ii) alone would not avoid the need for mid-circuit reset or repeated stabiliser operations.

[0077] Fig. 8 shows a more general quantum circuit for performing stabiliser measurement with two sets of stabiliser operations (e.g. X-type and Z-type stabiliser operations). Data qubit and syndrome qubit labels in Fig. 8 correspond to those used in Figs. 5 and 6. First and second unitary stages in a timestep t are labelled t / x(t) and t / 2(t) respectively (corresponding to the first unitary stage 502a and second unitary stage 502b of Fig. 5), and first and second measurement stages of timestep t are labelled M1(t) and M2(t) respectively (corresponding to the first measurement stage 504a and second measurement stage 504b of Fig. 5). t denotes an overall timestep in the syndrome extraction process. Fig. 8 also shows parts of other first and second unitary stages and t / 2(t) respectively from neighbouring stabiliser operations which act upon some of the same data qubits as the first and second unitary stages [^(t) and U2(t), as well as parts of unitary and measurement stages associated with preceding and subsequent timesteps.

[0078] The staggered nature of the timesteps is shown in Fig. 8: the first measurement stage M1(t) in timestep t is measuring the syndrome qubit states resulting from the second unitary stage U2(t - 1) in the preceding timestep. One skilled in the art will appreciate that a stabiliser measurement round could be defined as including all operations during a given overall timestep t (i.e. ^(t), M2(t), M1(t), U2(t) and corresponding operations for other stabilisers, such as and U2(ty), or it could alternatively be defined as in terms of the measurement stage and associated unitary stages (e.g. a stabiliser measurement round could be defined as encompassing U2(t - 1), ^(t), M1(t) and M2(t), along with corresponding operations for other stabiliser (equivalently, one could consider that the stabiliser measurement rounds for different sets of stabiliser operations (e.g. X-type and Z-type stabiliser operations) are offset slightly, i.e. different timestep labels could be used for different sets of stabiliser operations).

[0079] A stabiliser measurement round for a given stabiliser involves a single measurement stage (which could, for example, be a measurement of a single syndrome qubit in conventional methods, or parallel measurement of multiple syndrome qubits in the method disclosed herein); previous and subsequent measurement stages will represent different stabiliser measurement rounds associated with that stabiliser.

[0080] It should be understood that alternative quantum circuits may use additional syndrome qubits (i.e. three or more) for each stabiliser operation. In addition, alternative QEC codes may use circuits involving more or fewer data qubits as required. Furthermore, while the illustrated quantum circuits use two sets of stabiliser operations (i.e. a set of X-type stabiliser operations and a set of Z-type stabiliser operations), additional sets of stabiliser operations may also be utilised in some QEC codes (e.g. Y-type stabiliser operations and / or combinations of different Pauli operators), or the stabiliser operators may be divided into a larger number of sets.

[0081] Fig. 9 shows a method for performing a quantum error correction code involving multiple stabiliser measurement rounds using the stabiliser measurement techniques described herein. The method may be performed by a quantum computing system comprising a register of quantum devices, a control system coupled to the register of quantum devices, and a decoding system.

[0082] In a first step 902, the control system performs a plurality of stabiliser operations on the register of quantum devices to obtain syndrome data representative of an error state of the register of quantum devices. As described above, each stabiliser operation is associated with a stabiliser of the quantum error correction code and involves a unitary stage and a measurement stage during which multiple auxiliary quantum devices in the register of quantum devices are measured to provide multiple independent outcomes for the stabiliser associated with the stabiliser operation during each stabiliser measurement round. As also described above, the plurality of stabiliser operations involves multiple of sets of stabiliser operations (preferably non-overlapping sets), and measurement stages of stabiliser operations in each set are performed in parallel with unitary stages of stabiliser operations in a different set.

[0083] The syndrome data is then received at the decoding system in step 904 (it may either be received via the control system or directly from the register of qubits), and the decoder decodes the syndrome data in step 906 to determine a correction for the error state.

[0084] One skilled in the art will appreciate that numerous decoding algorithms can be used depending upon performance requirements and available computing resources. Suitable decoding algorithms include clustering decoders (such as union-find), minimum-weight- perfect-matching, belief propagation techniques etc. The additional stabiliser measurement values obtained using the methods disclosed herein provide extra syndrome data compared to conventional stabiliser measurement techniques, and decoding algorithms can use this additional syndrome data to provide improved decoding performance. For example, in decoding methods that utilise a decoding graph (or decoding hypergraph), syndrome data derived from each stabiliser measurement may be associated with one or more nodes of the decoding graph (for example, additional nodes and associated edges may be added to the decoding graph for each additional independent stabiliser outcome).

[0085] In addition, one skilled in the art will appreciate that it is generally not necessary to determine error locations and / or apply corrections to the qubits: it suffices to track (classically) how the encoded logical state (or states) of the qubits is affected by errors. The method may optionally further comprise measuring a logical state encoded in the quantum devices and applying the correction to the measured logical state.

[0086] Pauli X, Y and Z operations may be referred to herein simply as X, Y and Z operations.

[0087] Unless indicated otherwise, the accompanying figures are not necessarily to scale. For example, the durations of gates and measurements in the illustrated quantum circuit diagrams are not intended to be to scale.

[0088] While the present disclosure uses the convention that the qubit measurement outcomes associated with |0) and |1) quantum states are referred to as 0 and 1 respectively (i.e. referring to the states themselves), one skilled in the art will appreciate that the disclosure is equally applicable when using the alternative nomenclature in which these measurement outcomes are referred to as +1 and -1 respectively (i.e. the eigenvalues associated with the |0) and |1) states respectively). Any method described herein may be provided as a computer program product and / or on a computer readable medium such as a non-transitory computer readable medium.

[0089] It should be understood that any method of the present disclosure could include additional steps, and any device could include additional components. In addition, unless indicated otherwise or technically infeasible, the method steps disclosed herein may be performed in alternative orders, and any order described herein should be considered as exemplary rather than limiting.

[0090] Furthermore, one skilled in the art will appreciate that any computation that can be performed by a classical processing device can also be performed by a quantum computing device. Accordingly, any methods or described herein that is performed on a classical computing device (such as a CPU) can also be performed by a quantum processing device, such as a quantum processing unit (QPU) comprising a plurality of qubits.

Claims

CLAIMS1. A quantum computing system configured to perform a quantum error correction code involving multiple stabiliser measurement rounds, the quantum computing system comprising: a register of quantum devices; a control system coupled to the register of quantum devices; and a decoding system, wherein the control system is configured to perform a plurality of stabiliser operations on the register of quantum devices to obtain syndrome data representative of an error state of the register of quantum devices, wherein each stabiliser operation is associated with a stabiliser of the quantum error correction code and involves: a unitary stage; and, a measurement stage in which multiple auxiliary quantum devices in the register of quantum devices are measured to obtain multiple independent outcomes for the stabiliser associated with the stabiliser operation during each stabiliser measurement round, wherein the plurality of stabiliser operations comprises a plurality of sets of stabiliser operations, wherein measurement stages of stabiliser operations in each set are performed in parallel with unitary stages of stabiliser operations in a different set, and wherein the decoding system is configured to: receive the syndrome data; and decode the syndrome data to determine a correction for the error state.

2. The quantum computing system of claim 1 , wherein the quantum computing system is further configured to: measure a logical state encoded in the register of quantum devices to obtain a logical state measurement; and apply the correction to the logical state measurement.

3. The quantum computing system of any preceding claim, wherein the plurality of sets of stabiliser operations comprises a set of X-type stabiliser operations and a set of Z-type stabiliser operations.

4. The quantum computing system of any preceding claim, wherein each independent outcome is obtained from measurement of a single auxiliary quantum device.

5. The quantum computing system of any preceding claim, wherein each independent outcome is obtained from measurement of multiple auxiliary quantum devices.

6. The quantum computing system of any preceding claim, wherein each unitary stage comprises a plurality of entangling gates.

7. The quantum computing system of any preceding claim, wherein the quantum devices are qubits.

8. A method of performing a quantum error correction code involving multiple stabiliser measurement rounds, the method performed at a quantum computing system comprising a register of quantum devices, a control system coupled to the register of quantum devices, and a decoding system, the method comprising: performing, by the control system, a plurality of stabiliser operations on the register of quantum devices to obtain syndrome data representative of an error state of the register of quantum devices, receiving, at the decoding system, the syndrome data; and decoding, at the decoding system, the syndrome data to determine a correction for the error state, wherein each stabiliser operation is associated with a stabiliser of the quantum error correction code and involves: a unitary stage; and, a measurement stage in which multiple auxiliary quantum devices in the register of quantum devices are measured to obtain multiple independent outcomes for the stabiliser associated with the stabiliser operation during each stabiliser measurement round, and wherein the plurality of stabiliser operations comprises a plurality of sets of stabiliser operations, wherein measurement stages of stabiliser operations in each set are performed in parallel with unitary stages of stabiliser operations in a different set.

9. The method of claim 8, further comprising: measuring a logical state encoded in the register of quantum devices to obtain a logical state measurement; and applying the correction to the logical state measurement.

10. The method of claim 8 or claim 9, wherein the plurality of sets of stabiliser operations comprises a set of X-type stabiliser operations and a set of Z-type stabiliser operations.

11. The method of any of claims 8 to 10, wherein each independent outcome is associated with a single measurement value.

12. The method of any of claims 8 to 11 , wherein each independent outcome is associated with multiple measurement values.

13. The method of any of claims 8 to 12, wherein each unitary stage comprises a plurality of entangling gates.

14. The method of any of claims 8 to 13, wherein the quantum devices are qubits.

15. A computer-readable medium comprising instructions which, when executed by a quantum computing system, cause the quantum computing system to carry out the method of any of claims 8 to 14.

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