Method for performing logical qubit operations

Abstract

Methods, quantum computing systems, systems, and computer-readable media are provided for synchronising qubits during a logical qubit operation. The logical qubit operation is executed according to a synchronisation scheme which defines synchronisation groups comprising partially overlapping sets of qubits for different time periods of the logical qubit operation, where the synchronisation groups form a logical chain of partially overlapping synchronisation groups between two logical qubits, to reduce the number of physical qubits which must be synchronised with one another.

Description

METHOD FOR PERFORMING LOGICAL QUBIT OPERATIONSField of the Invention

[0001] The present invention relates generally to quantum computing systems, more specifically to methods, systems, computer-readable media for maintaining synchronisation between data qubits during a logical qubit operation.Background

[0002] Quantum computing systems, often referred to as quantum computers, utilise quantum mechanical phenomena to perform computational tasks. Quantum computing systems are formed of many quantum devices, for example qubits, which may be operated on by quantum gates to manipulate the state of the qubit to perform a particular calculation. Generally, quantum computers are operated with a high degree of redundancy, such that multiple physical qubits collectively implement a single logical qubit. This minimises the effects of physical qubit errors in the state of logical qubits, however this increases the total number of physical qubits required for the quantum computing system.

[0003] Providers of quantum computing systems are continually striving to increase the number of logical qubits in their quantum computing systems, requiring a significant increase in the number of physical qubits required to encode the state of these logical qubits. However, operating a quantum computing system with large numbers of qubits can present considerable challenges, particularly when executing logical qubit operations between multiple logical qubits.Summary of the Invention

[0004] The invention is defined by the appended claims.

[0005] According to a first aspect, there is provided a method for maintaining synchronisation between data qubits during a logical qubit operation between a plurality of logical qubits implemented by a quantum computing system, wherein the quantum computing system includes a plurality of physical qubits implementing the plurality of logical qubits, wherein the plurality of physical qubits includes a plurality of physical qubit subsets, wherein the method comprises: receiving an indication of: a logical qubit operation between a first logical qubit and a second logical qubit to be executed by the quantum computing system, and a synchronisation scheme, wherein the first logical qubit is implemented by a first physical qubit subset of the quantum computing system, a second logical qubit is implemented by a second physical qubit subset of the quantum computing system, and wherein the plurality of physical qubits of the quantum computing system further includes one or more additional physical qubit subsets, wherein the synchronisation scheme defines a plurality of time periods of the logical qubit operation and one or more synchronisation groups for each of the plurality of time periods, wherein each of the synchronisation groups includes two or more physical qubit subsets to be synchronised with one another for a respective time period, and wherein the synchronisation groups partially overlap with one another to form a logical chain of synchronisation groups between the first and second physical qubit subsets spanning the plurality of time periods; and executing the logical qubit operation by: in an initial time period, performing one or more physical qubit operations on physical qubits within one or more initial synchronisation groups for the initial time period defined by the synchronisation scheme; and in a later time period subsequent to the first time period, performing one or more physical qubit operations on physical qubits within one or more subsequent synchronisation groups for the later time period defined by the synchronisation scheme.

[0006] As such, according to the invention, a logical qubit operation between two logical qubits may be performed in a manner which minimises the synchronisation requirements for physical qubits utilised in performing the logical qubit operation. That is, by defining a logical chain of partially overlapping synchronisation groups which span multiple time periods, smaller groups of physical qubits are synchronised with one another during the logical qubit operation. Accordingly, performance of the logical qubit operation can be significantly simplified, as synchronisation of physical qubits can be accurately achieved more easily, while only slightly increasing the overall time over which the logical qubit operation is executed.

[0007] In addition, as the number of physical qubits which are synchronised at any given time during the logical qubit operation is reduced, control systems for the quantum computing systems may have reduced complexity and / or cost. Moreover, decoding of syndrome data from the quantum computing system may be simplified, as syndrome data from fewer physical qubits may be received and decoded within a given time period (and as such the use of spatial windowing techniques to decode syndrome data may not be required to decode syndrome data). Accordingly, the decoding system from the quantum computing system may also have reduced complexity and / or cost.

[0008] The synchronisation scheme may be thought of as a definition of which physical qubit subsets are to be synchronised with one another at different time periods of the logical qubit operation. As such, the synchronisation scheme may define (i.e. include) the number of time periods of the logical qubit operation, the number of synchronisation groups (e.g. the number of initial synchronisation groups, the number of later synchronisation groups, and the number of any other synchronisation groups), and / or the physical qubit subsets included within each synchronisation group.

[0009] A synchronisation group may be thought of as a group of physical qubits (e.g. a group comprising a plurality of physical qubit subsets) which are to be synchronised with one another for a given time period during the logical qubit operation. In other words, in a given time period, all physical qubit subsets which are to be synchronised with one another for the given time period are in the same synchronisation group. Multiple synchronisation groups may exist for a single time period of the logical qubit operation, such that physical qubit subsets within a given synchronisation group for a given time period are to be synchronised with one another, but not with physical qubit subsets within a different synchronisation group for the given time period.

[0010] The synchronisation groups may be thought of as partially overlapping with one another such that some but not all of the physical qubit subsets of a synchronisation group overlap with some but not all of the physical qubit subsets of another synchronisation group of another time period. Furthermore, each synchronisation group overlaps with one or more other synchronisation groups for one or more other time periods of the logical qubit operation. As such, partially overlapping synchronisation groups extending from the first physical qubit subset to the second physical qubit subset and spanning multiple time periods of the logical qubit operation form the logical chain of synchronisation groups. In some cases, each synchronisation group comprises at least three physical qubit subsets, where, for a first synchronisation group for a first time period, a first physical qubit subset is also included in a second synchronisation group for a second time period; a second physical qubit subset either: is included in a third synchronisation group for the first, or a third time period or implements a logical qubit of the logical qubit operation; and one or more third physical qubit subsets that are unique to the first synchronisation group (i.e. is not included in any other synchronisation groups) and do not implement a logical qubit of the logical qubit operation.

[0011] The physical qubit subsets may be subsets (i.e. groups including some but not all) of the plurality of physical qubits of the quantum computing system. The physical qubit subsets implementing (i.e. encoding) the logical qubits may be distinct from (i.e. separate from) one another. Furthermore, a physical qubit subset may implement a logical qubit such that, at a given time, the state of a logical qubit is encoded by some or all of the physical qubits of the physical qubit subset. It should be appreciated, that the exact physical qubits of the physical qubit subset implementing the logical qubit may change, including during performance of the logical qubit operation.

[0012] The physical qubit subsets may have particular sizes, which may e.g. be defined in terms of code distance. As understood by a person skilled in the art, code distance may be defined as the minimum number of physical errors required for an undetectable logical error, such that quantum error correction codes can generally reliably correct errors spanning less than half the code distance. As such, each of the physical qubit subsets may include at least a number of physical qubits to provide a predetermined code distance “d” when encoded on a surface code or other quantum error correcting code. Therefore, each physical qubit subset may have the same predetermined code distance “d”, or the code distance may be different between physical qubit subsets, but may be at least the predetermined code distance “d”. For example, each physical qubit subset may include at least a same number of physical qubits as the first and second physical qubit subsets. It should be appreciated that the size of each physical qubit subset may be chosen by the person skilled in the art so as to ensure that the code distance is maintained at all times for any given schedule, and that the actual size of each physical qubit subset will depend upon various factors, such as the gate (i.e. physical qubit manipulation) schedules being used and the quantum error-correction (QEC) code being implemented, as would be understood by the person skilled in the art.

[0013] In addition to the physical qubit subsets which implement the logical qubits, the quantum computing system includes the one or more additional physical qubit subsets. The one or more additional physical qubit subsets may include physical qubits not included in the physical qubit subsets implementing the plurality of logical qubits. In other words, some or all of the physical qubits included in the one or more additional physical qubit subsets may be distinct from the physical qubit subsets which implement the logical qubits.

[0014] The logical qubit operation is a logical operation between different logical qubits encoded by different physical qubit subsets. The logical qubit operation may be of various types. For example, the logical qubit operation may be an entangling operation to entangle the logical qubits. In some cases, the logical qubit operation may be a lattice surgery operation executed on logical qubits encoded by physical qubit subsets of a surface code. According to some examples, the logical qubit operation may comprise a relocation of the first logical qubit and / orthe second logical qubit. It should be understood that relocation of a logical qubit refers to the relocation of the logical qubit state from one physical qubit subset to another physical qubit subset. That is, the physical qubits themselves are not relocated, but rather the state of the physical qubit subsets is modified such that the logical state corresponding to a logical qubit is relocated from one physical qubit subset to another. It should be appreciated that any relocation of the logical qubits may include a relocation of the first logical qubit and / or the second logical qubit to different physical qubits. In some cases, one of the logical qubits may be relocated relative to one or more other logical qubits. It should be appreciated that any relocation of the logical qubit(s) may be the logical qubit operation, or may form part of a larger qubit operation (e.g. an entangling operation). For example, one or more of the logical qubits may be relocated before being entangled with one or more otherlogical qubits. However, it should also be appreciated that logical qubits may in some cases be entangled without relocating any of the logical qubits.

[0015] According to some examples, the one or more initial synchronisation groups includes the first qubit subset, and wherein the one or more later synchronisation groups include the second qubit subset. As such, the logical qubits may form part of synchronisation groups for different time periods of the logical qubit operation. Accordingly, flexibility is provided in the manner in which synchronisation is maintained during the logical qubit operation.

[0016] According to some examples, the one or more initial synchronisation groups include: a first initial synchronisation group comprising the first qubit subset, and one or more other second initial synchronisation groups comprising additional physical qubit subsets, wherein the one or more later synchronisation groups include: a first later synchronisation group comprising the second qubit subset, and one or more other later synchronisation groups, wherein the one or more other second initial synchronisation groups partially overlap with the first later synchronisation group and the one or more other later synchronisation groups, and wherein the one or more other later synchronisation groups partially overlap with the first initial synchronisation group and the one or more other initial synchronisation groups. As such, the logical qubit operation may be performed using long logical chains of synchronisation groups for the logical qubit operation without requiring that the first and second physical qubit subsets are synchronised within their respective synchronisation groups during the same time period, thereby providing flexibility in the manner in which synchronisation is maintained during the logical qubit operation.

[0017] According to some examples, the one or more initial synchronisation groups includes a first initial synchronisation group and a second initial synchronisation group, wherein the first initial synchronisation group includes the first qubit subset, and wherein the second initial synchronisation group includes the second qubit subset, and wherein the one or more later synchronisation groups partially overlap with the first initial synchronisation group and the second initial synchronisation group. Alternatively, the one or more later synchronisation groups includes a first later synchronisation group and a second later synchronisation group, wherein the first later synchronisation group includes the first qubit subset, and wherein the second later synchronisation group includes the second qubit subset, and wherein the one or more initial synchronisation groups partially overlap with the first later synchronisation group and the second later synchronisation group. As such, the logical chain of synchronisation groups may be defined in a variety of different ways, thereby providing flexibility in the manner in which synchronisation is maintained during the logical qubit operation.

[0018] According to some examples, the method further comprises: in a further time period between the initial and later time periods, performing one or more physical qubit operations on physical qubits within one or more third synchronisation groups for the third time period defined by the synchronisation scheme, wherein the one or more third synchronisation groups partially overlap the one or more first synchronisation groups and / or the one or more second synchronisation groups. As such, the logical qubit operation may be performed using long logical chains of synchronisation groups spanning three or more time periods. Accordingly, flexibility is provided in the manner in which synchronisation is maintained during the logical qubit operation.

[0019] According to some examples, each physical qubit subset comprises a plurality of data qubits and a plurality of syndrome qubits. Accordingly, errors in the state of the logical qubit encoded by the data qubits may be identified and corrected.

[0020] According to some examples, the synchronisation groups may partially overlap with one another such that a number of physical qubits common to the partially overlapping synchronisation groups is at least a same number of physical qubits as the first and second physical qubit subsets. In other words, the size of the overlap between synchronisation groups may be at least the predetermined code distance “d”. As such, errors in the logical chain of synchronisation groups can be readily identified and corrected.

[0021] According to some examples, each synchronisation group comprises physical qubit subsets for which the quantum computing system is capable of performing multi-qubit operations across the physical qubit subsets within the respective time period for the synchronisation group. As such, the logical chain of synchronisation groups between the logical qubits may be implemented using the additional qubit subsets in a manner which allows the logical qubit operation between the logical qubit operation to be executed.

[0022] According to some examples, the method further comprises: prior to performing physical qubit operations in a particular time period, synchronising the physical qubit subsets of each synchronisation group for the respective time period. As such, synchronisation between physical qubits (to within a predetermined margin of error) may be maintained within the synchronisation groups. In some cases, synchronising the physical qubit subsets comprises synchronising control hardware for performing physical qubit operations on the physical qubits of the respective physical qubit subsets. As such, physical qubit operations may be performed on physical qubits within a synchronisation group in a synchronised manner (i.e. simultaneously, to within a predetermined margin of error). The control hardware may, for example, include hardware for generating laser pulses for manipulating physical qubits. Synchronisation of the control hardware may therefore include synchronising the generation of laser pulses by the control hardware. It should, however, be appreciated that the control hardware is not limited to hardware for generating laser pulses, and that the invention is compatible with other control hardware.

[0023] According to some examples, the indication of the synchronisation scheme is the indication of the logical qubit operation; and the method further comprises deriving the synchronisation scheme based on the indication of the logical qubit operation. In other words, no explicit indication of the synchronisation scheme is received. Instead, an indication of the logical qubit operation is received and the synchronisation scheme is derived based on the indication of the logical qubit operation. For example, the synchronisation scheme may be derived based on the type of logical qubit operation, the logical qubits forming part of the logical qubit operation, the physical qubit subsets (e.g. the size of the physical qubit subsets) available for the logical qubit operation, a number of physical qubits included in the one or more additional qubit subsets, synchronisation capabilities of the control hardware of the quantum computing system, and / or one or more other factors. In this way, the amount of data the quantum computing system receives in order to identify the synchronisation scheme is minimised.

[0024] According to a second aspect of the invention, there is provided a quantum computing system comprising: a plurality of physical qubits configured to implement a plurality of logical qubits, wherein the plurality of physical qubits includes a plurality of physical qubit subsets, wherein a first logical qubit is implemented by a first physical qubit subset, a second logical qubit is implemented by a second physical qubit subset, and wherein the plurality of physical qubits further includes one or more additional physical qubits; and a control system configured to perform physical qubit operations on the plurality of physical qubits to execute logical qubit operations; wherein the quantum computing system is configured to perform the method as described above.

[0025] According to a third aspect of the invention, there is provided a computer-readable medium comprising instructions which, when executed by the quantum computing system described above, cause the quantum computing system to perform the method as also described above.

[0026] According to a fourth aspect of the invention, there is provided a computer-implemented method comprising: identifying a logical qubit operation to be performed on a plurality of logical qubits and a quantum computing system on which the logical qubit operation is to be performed; based on the identified logical qubit operation and quantum computing system, identifying a synchronisation scheme for the logical qubit operation, wherein the synchronisation scheme defines a plurality of time periods of the logical qubit operation and synchronisation groups for each of the plurality of time periods, wherein each of the synchronisation groups includes two or more physical qubit subsets of the quantum computing system to be synchronised with one another for a respective time period, and wherein the synchronisation groups partially overlap with one another to form a logical chain of synchronisation groups between physical qubit subsets implementing the plurality of logical qubits, wherein the logical chain of synchronisation groups spans the plurality of time periods; and transmitting an indication of the logical qubit operation and the synchronisation scheme to the quantum computing system.

[0027] As such, according to the invention, a logical qubit operation between two logical qubits may be signalled to a quantum computing system such that the logical qubit operation may be performed in a manner which minimises the synchronisation requirements for physical qubits utilised in performing the logical qubit operation. That is, by defining a logical chain of partially overlapping synchronisation groups which span multiple time periods, smaller groups of physical qubits are synchronised with one another during the logical qubit operation. Accordingly, performance of the logical qubit operation can be significantly simplified, as synchronisation of physical qubits can be accurately achieved more easily, while only slightly increasing the overall time over which the logical qubit operation is executed.

[0028] According to some examples, identifying the synchronisation scheme is based on a code distance of the plurality of logical qubits. That is, the physical qubit subsets and synchronisation groups may be configured such that the code distance of the plurality of logical qubits (if the logical qubits have different code distances, this may be the minimum code distance of all logical qubits involved in the logical qubit operation) is maintained throughout the synchronisation scheme; one skilled in the art will appreciate that the details of how the code distance is maintained will vary depending upon factors such as (among others) the arrangement of the physical qubits, the logical qubit operation being performed, and the type of quantum error correction code being used. Furthermore, the synchronisation scheme may be additionally or alternatively derived based on the type of logical qubit operation, the logical qubits forming part of the logical qubit operation, the physical qubit subsets (e.g. the size of the physical qubit subset) available for the logical qubit operation, synchronisation capabilities of the control hardware of the quantum computing system, and / or one or more other factors. As such, the synchronisation scheme may be bespoke to each logical qubit operation, thereby providing flexibility and performance maximisation.

[0029] According to some examples, the method further comprises: transmitting an indication of the logical qubit operation and the synchronisation scheme to a decoder for the quantum computing system. As such, both the quantum computing system and a decoder for the quantum computing system can operate according to the same synchronisation scheme for the logical qubit operation. Therefore, not only can the logical qubit operation be accurately performed, bydata (e.g. readout data and / or syndrome data) can be accurately read from the quantum computing system.

[0030] According to a fifth aspect of the invention, there is provided a computer-implemented method comprising: receiving an indication of a logical qubit operation to be performed on a quantum computing system and a synchronisation scheme, wherein the synchronisation scheme defines a plurality of time periods of the logical qubit operation for a plurality of logical qubits implemented by the quantum computing system and synchronisation groups for each of the plurality of time periods, wherein each of the synchronisation groups includes two or more physical qubit subsets of the quantum computing system to be synchronised with one another for a respective time period, and wherein the synchronisation groups partially overlap with one another to form a logical chain of synchronisation groups between physical qubit subsets implementing the plurality of logical qubits, wherein the logical chain of synchronisation groups spans the plurality of time periods; and receiving syndrome data from the quantum computing system for the logical qubit operation, wherein receiving syndrome data comprises: in an initial time period, receiving initial syndrome data for physical qubits within one or more initial synchronisation groups for the initial time period defined by the synchronisation scheme; and in a later time period subsequent to the first time period, receiving subsequent syndrome data for physical qubits within one or more subsequent synchronisation groups for the later time period defined by the synchronisation scheme.

[0031] As such, according to the invention, syndrome data may be received from a quantum computing system for a logical qubit operation between two logical qubits in a manner consistent with the synchronisation scheme used by the quantum computing system for performing the logical qubit operation. As such, syndrome data can be decoded accurately to identify errors in the logical qubit operation.

[0032] In some cases, decoding syndrome data may comprise determining a correction to the state of one or more physical qubits of the quantum computing system. Syndrome data may be thought of as data indicative of an error in the state of one or more physical qubits. Syndrome data may therefore be in the form of readout data output by the quantum computing system.

[0033] According to some examples, the method further comprises: upon receiving the initial syndrome data, decoding the initial syndrome data; and upon receiving the subsequent syndrome data, decoding the subsequent syndrome data. In other words, the decoder may begin decoding the syndrome data for each time period defined in the synchronisation scheme upon receiving said syndrome data (i.e. before the logical qubit operation has finished executing), thereby minimising the additional time taken to decode the syndrome data. Alternatively, the method further comprises: based on receiving both the initial syndrome data and subsequent syndrome data, decoding the initial syndrome data and subsequent syndrome data. In other words, the decoder may wait until the logical qubit operation has concluded before decoding the syndrome data.

[0034] According to a sixth aspect of the invention, there is provided a system comprising: a memory; and one or more processors, wherein the one or more processors are configured to perform any of the methods described above.

[0035] According to a seventh aspect of the invention, there is provided a computer-readable medium comprising instructions which, when executed by the computer, cause the computer to perform any of the methods described above.Brief Description of the Drawings

[0036] Embodiments of the invention will now be described, by way of example only, with reference to the following figures.

[0037] Figure 1 illustrates an entangling operation between two logical qubits.

[0038] Figure 2 illustrates a synchronisation scheme for a logical qubit operation between two logical qubits according to an example of the present disclosure.

[0039] Figure 3 illustrates a synchronisation scheme for a logical qubit operation between two logical qubits according to an example of the present disclosure.

[0040] Figure 4 illustrates a synchronisation scheme for a logical qubit operation between two logical qubits according to an example of the present disclosure.

[0041] Figure 5 illustrates a synchronisation scheme for a logical qubit operation between two logical qubits according to an example of the present disclosure.

[0042] Figure 6 illustrates a synchronisation scheme for a logical qubit operation between two logical qubits according to an example of the present disclosure.

[0043] Figure 7 illustrates a synchronisation scheme for a logical qubit operation between two logical qubits according to an example of the present disclosure.

[0044] Figure 8 illustrates a synchronisation scheme for a relocation operation according to an example of the present disclosure.

[0045] Figure 9 illustrates a synchronisation scheme for a logical qubit operation between two logical qubits according to an example of the present disclosure.

[0046] Figures 10A-D illustrate a topological surface code view of an entangling operation according to the synchronisation scheme shown in Figure 4.

[0047] Figure 11 illustrates a system capable for implementing examples of the present disclosure.

[0048] Figure 12 illustrates a computing system for implementing examples of the present disclosure.

[0049] Figure 13 illustrates a flowchart for a method according to an example teaching of the present disclosure.

[0050] Figure 14 illustrates a flowchart for a method according to an example teaching of the present disclosure.

[0051] Figure 15 illustrates a flowchart for a method according to an example teaching of the present disclosure.

[0052] Any reference to prior art documents in this specification is not to be considered an admission that such prior art is widely known or forms part of the common general knowledge in the field. As used in this specification, the words “comprises”, “comprising”, and similar words, are not to be interpreted in an exclusive or exhaustive sense. In other words, they are intended to mean “including, but not limited to”. The invention is further described with reference to the following examples. It will be appreciated that the invention as claimed is not intended to be limited in any way by these examples. It will also be recognised that the invention covers not only individual embodiments but also combination of the embodiments described herein.

[0053] The various embodiments described herein are presented only to assist in understanding and teaching the claimed features. These embodiments are provided as a representative sample of embodiments only, and are not exhaustive and / or exclusive. It is to be understood that advantages, embodiments, examples, functions, features, structures, and / or other aspects described herein are not to be considered limitations on the scope of the invention as defined by the claims or limitations on equivalents to the claims, and that other embodiments may be utilised and modifications may be made without departing from the spirit and scope of the claimed invention. Various embodiments of the invention may suitably comprise, consist of, or consist essentially of, appropriate combinations of the disclosed elements, components, features, parts, steps, means, etc., other than those specifically described herein. In addition, this disclosure may include other inventions not presently claimed, but which may be claimed in future.Detailed Description

[0054] As the numbers of logical qubits increases in quantum computing systems, so too does the number of physical qubits encoding the state of those logical qubits. This can present many challenges, in particularwhen performing logical qubit operations involving multiple logical qubits. When performing logical qubit operations, such as an entangling operation, the physical qubits utilised in executing the logical qubit operation should be synchronised with one another such that the manipulation of the physical qubits by control hardware is synchronised for the duration of the logical qubit operation. In this way, manipulations of physical qubits as part of the logical qubit operation may be performed substantially simultaneously (within a predetermined margin of error).

[0055] Figure 1 illustrates an entangling operation between two logical qubits 110 executed on a quantum computing system having a plurality of physical qubits according to conventional approaches. Figure 1 illustrates the logical qubits 110 as being implemented by respective sets of physical qubits 120, where the sets of physical qubits 120 are shown in one-dimension (extending along the x (i.e. left-right) axis) where the x axis corresponds to a size of physical qubits. Figure 1 illustrates a first qubit subset 120(1) which implements (i.e. collectively encodes the state of) a first logical qubit 110(1), and a second qubit subset 120(3) which implements a second logical qubit 110(2). In this example, the first physical qubit subset 120(1) has a size d, the second physical qubit subset 120(5) also has a size d. Additionally, the quantum computing system includes additional physical qubits 120(2) having size 3d, where the additional physical qubits 120(2) are shown between the first and second physical qubit subsets 120(1), 120(3). It should be appreciated that, in certain quantum computing systems, it may be necessary (or desirable) to use the additional physical qubits 120(2) to entangle the first and second logical qubits 110(1), (2), as in Figure 1. It should also be appreciated that the illustration of the additional physical qubits 120(2) between the first and second physical qubit subsets 120(1), 120(3) does not necessarily mean that the additional physical qubits 120(2) are located physically between the first and second physical qubit subsets 120(1), 120(3). Rather, this illustration, as will be explained below, illustrates that the additional physical qubits 120(2) are utilised in the entangling operation between the logical qubits implemented by the first and second physical qubit subsets 120(1), 120(3).

[0056] In Figure 1 , time is shown on the y (up-down) axis, with time increasing upwards in the diagram. Physical qubits which are synchronised with one another at any given time are shown between solid lines in Figure 1. For example, at time to, before the logical qubit operation has started execution, qubits within the physical qubit subset 120(1) which implement the first logical qubit 110(1) are synchronised with one another and as such are shown between two solid verticallines. Similarly, at time to, qubits within the physical qubit subset 120(3) which implement the second logical qubit 110(2) are synchronised with one another and as such are shown between two solid vertical lines. However, the additional physical qubits 120(2) are not synchronised with one another, or with the first and / or second physical qubit subsets 120(1), 120(3).

[0057] At time ti, execution of the entangling operation begins and continues until time t2. Various physical qubit manipulations are executed between times ti and t2in order to entangle the first and second logical qubits 110(1), (2). As such, the first physical qubit subset 120(1), second physical qubit subset 120(3), and additional physical qubits 120(2) should be synchronised with one another during the entangling operation between times ti and t2, such that the physical qubit manipulations are executed in a synchronised manner. That is, the physical qubit manipulations may include one or more layers of physical qubit manipulations (i.e. gates), and as such the physical qubit manipulations are executed in a synchronised manner such that each physical qubit manipulation in a given layer for any is executed in a synchronised manner (i.e. simultaneously to within a predefined margin of error). Accordingly, between times ti and t2all of the first physical qubit subset 120(1), second physical qubit subset 120(3), and additional physical qubits 120(2) are illustrated as being located between the outermost solid vertical lines, and the additional physical qubits 120(2) are shown between horizontal solid lines. In other words, the first physical qubit subset 120(1), second physical qubit subset 120(3), and additional physical qubits 120(2) are illustrated as being connected to one another (without solid lines between them).

[0058] Before or at time t2, a measurement of physical qubits within additional physical qubits 120(2) is performed to complete the entangling operation. After time t2the first and second logical qubits 110(1), (2) have become entangled with one another, and as such the first physical qubit subset 120(1), second physical qubit subset 120(3), and additional physical qubits 120(2) no longer need to be synchronised with one another. Accordingly, from time t2onwards the first and second logical qubits 110(1), (2) are illustrated as being between separate solid vertical lines, as the physical qubits of the first physical qubit subset 120(1) remain synchronised with one another, and the physical qubits of the second physical qubit subset 120(3) also remain synchronised with one another.

[0059] Using this conventional approach, two logical qubits can theoretically be entangled with one another. However, this approach presents practical challenges as it requires synchronisation between all of the first physical qubit subset 120(1), second physical qubit subset 120(3), and additional physical qubits 120(2), which in the present example spans a distance of 5d. Depending on the distance between the first physical qubit subset 120(1) and second physical qubit subset 120(3), this can be difficult to implement in reality, as potentially vast numbers of qubits require collective synchronisation.

[0060] In particular, numerous classical (i.e. non-quantum) components are required in order to control large quantum computing systems, each having their own internal clock. As such, when performing logical qubit operations, the clocks of these classical components should be synchronised and kept synchronised (i.e. with no drift) during the logical qubit operation. Furthermore, certain signals (i.e. control signals or messages) within the quantum computing system should be timed to cause those signals to arrive at their respective destinations at the same time. While this can in principle be achieved on some quantum computing systems, doing so can increase the complexity in performing a quantum operation, and increase the time taken to perform the operation. For example, components may need to be re-synchronised during a quantum operation, or artificial delays may need to be added to maintain synchronisation.

[0061] According to the present invention, a new approach for maintaining synchronisation between data qubits during a logical qubit operation between a plurality of logical qubits is provided. The approach includes receiving an indication of: a logical qubit operation between a first logical qubit and a second logical qubit to be executed by the quantum computing system, and a synchronisation scheme, wherein the first logical qubit is implemented by a first physical qubit subset of the quantum computing system, a second logical qubit is implemented by a second physical qubit subset of the quantum computing system, and wherein the plurality of physical qubits of the quantum computing system further includes one or more additional physical qubit subsets. The synchronisation scheme defines a plurality of time periods of the logical qubit operation and one or more synchronisation groups for each of the plurality of time periods, wherein each of the synchronisation groups includes two or more physical qubit subsets to be synchronised with one another for a respective time period, and wherein the synchronisation groups partially overlap with one another to form a logical chain of synchronisation groups between the first and second physical qubit subsets spanning the plurality of time periods. The logical qubit operation is then executed by: in an initial time period, performing one or more physical qubit operations on physical qubits within one or more initial synchronisation groups for the initial time period defined by the synchronisation scheme; and in a later time period subsequent to the first time period, performing one or more physical qubit operations on physical qubits within one or more subsequent synchronisation groups for the later time period defined by the synchronisation scheme.

[0062] Figure 2 illustrates an example implementation of a logical qubit operation according to this approach, where a logical qubit operation between a first logical qubit 210(1) and a second logical qubit 210(2) is performed. The first logical qubit 210(1) is implemented by a first physical qubit subset 220(1) and the second logical qubit 210(2) is implemented by a second physical qubit subset 220(5). The quantum computing system on which the logical qubit operation is performed also includes additional physical qubits, including three additional physical qubit subsets 220(2)-(4). In the present example, the first and second physical qubit subsets 220(1), (5) each have a size d. Additionally, the quantum computing system includes additional physical qubit subsets 220(2)-(4) each having size d, where the additional physical qubit subsets 220(2)-(4) are shown between the first and second physical qubit subsets 220(1), 220(5). As discussed in relation to Figure 1 , for certain quantum computing systems it may be either necessary or desirable to utilise the additional physical qubit subsets 220(2)-(4) in performing logical qubit operations between the first and second logical qubits 210(1), (2), for example due to the physical qubit connectivity capabilities of the quantum computing system. As such, the present disclosure relates to examples where a given number or set of additional physical qubits are to be utilised for performing logical qubit operations between logical qubits.

[0063] It should be appreciated that the illustration of the additional physical qubit subsets 220(2)- (4) between the first and second physical qubit subsets 220(1), 220(5) does not necessarily mean that the additional physical qubit subsets 220(2)-(4) are located physically between the first and second physical qubit subsets 220(1), 220(5). Rather, this illustration, as will be explained below, illustrates that the additional physical qubit subsets 220(2)-(4) are utilised in the logical qubit operation between the logical qubits implemented by the first and second physical qubit subsets 220(1), 220(5). This is true for each of the examples illustrated in the present disclosure. Furthermore, while each of the physical qubit subsets 220(1)-(5) are shown as having a size d, it should be appreciated that the physical qubit subsets may have different sizes. For example, the first and second physical qubit subsets 220(1), (5) may have the same size or different sizes. Furthermore, the additional physical qubit subsets 220(2)-(4) may have the same size or differentsizes. Moreover, the additional physical qubit subsets 220(2)-(4) may have the same size as or different sizes to the first and second physical qubit subsets 220(1), (5).

[0064] In the example of Figure 2, various physical qubits are synchronised with one another according to a synchronisation scheme. The synchronisation scheme may be thought of as a definition of which physical qubit subsets are to be synchronised with one another at different time periods of the logical qubit operation. As such, the synchronisation scheme may define (i.e. include) the number of time periods of the logical qubit operation, the number of synchronisation groups (e.g. the number of initial synchronisation groups, the number of later synchronisation groups, and the number of any other synchronisation groups), and / or the physical qubit subsets included within each synchronisation group.

[0065] In the present example, the synchronisation scheme defines that in a first time period 230(1) of the logical qubit operation between times ti and t2, physical qubits in an initial synchronisation group 240(1) (comprising physical qubit subsets 220(1)-(3)) are to be synchronised with one another. The synchronisation scheme also defines that in a second time period 230(2) of the logical qubit operation between times t2and t3, physical qubits in a later synchronisation group 240(2) (comprising physical qubit subsets 220(3)-(5)) are to be synchronised with one another. As can be seen from Figure 2, the initial and later synchronisation groups 240(1), (2) partially overlap one another such that the additional qubit subset 220(3) is included in both the initial and later synchronisation groups 240(1), (2). As such, a logical chain of partially overlapping synchronisation groups 240(1)-(2) is defined between the first and second physical qubit subsets 220(1), (5), spanning multiple time periods 240(1)-(2).

[0066] According to the example of Figure 2, the logical qubit operation is therefore spread over two time periods 230(1)-(2), and in each time period 230 fewer physical qubits are to be synchronised with one another than during the logical qubit operation in Figure 1. Therefore, although five additional physical qubit subsets 210(2)-(4) are utilised for performing the logical qubit operation, synchronisation across only three physical qubit subsets is implemented at any given time during the logical qubit operation. In this way, the number of physical qubits which should be synchronised with one another may be effectively capped, while still allowing execution of the logical qubit operation. Accordingly, by providing multiple time periods for the logical qubit operation and synchronisation groups of physical qubits for the different time periods, the synchronisation requirements for the logical qubit operation can be reduced as compared to the conventional approach required illustrated in Figure 1.

[0067] It should be appreciated that the example discussed in relation to Figure 2 is just one implementation of this general approach, and that other examples are envisaged. One such example implementation is shown in Figure 3. In the example of Figure 3, a logical qubit operation between a first logical qubit 310(1) and a second logical qubit 310(2) is performed. The first logical qubit 310(1) is implemented by a first physical qubit subset 320(1) and the second logical qubit 310(2) is implemented by a second physical qubit subset 320(7). The quantum computing system on which the logical qubit operation is performed also includes additional physical qubits, including five additional physical qubit subsets 320(2)-(6). In the present example, the first and second physical qubit subsets 320(1), (7) each have a size d, and the quantum computing system includes additional physical qubit subsets 320(2)-(6), where each of the additional physical qubit subsets 320(2)-(6) has a size d. Furthermore, while each of the physical qubit subsets 320(1)-(7) are shown as having a size d, it should be appreciated that the physical qubit subsets may have different sizes, as discussed in relation to Figure 2.

[0068] According to the synchronisation scheme for the example of Figure 3, the logical qubit operation is performed over three time periods 330. A first time period 330(1) occurs from time ti to t2; a second time period 220(2) occurs from time t2to t3; and a third time period 330(3) occurs from time t3to t4, after which the logical qubit operation has ended. In the first time period 330(1), there is an initial synchronisation group 340(1) comprising physical qubit subsets 320(1)-(3). In the second time period 330(2) there is a further synchronisation group 340(2) comprising physical qubit subsets 320(3)-(5). In the third time period 330(3) there is a later synchronisation group 340(3) comprising physical qubit subsets 320(5)-(7). As such, the three partially overlapping synchronisation groups 340(1 )-(3) form a logical chain of synchronisation groups between the first and second physical qubit subsets 320(1), (7) and spanning three time periods 330(1)-(3). Accordingly, the two logical qubits 310(1)-(2) can be synchronised with one another using five additional physical qubit subsets 320(2)-(6), while never requiring synchronisation between groups of more than three physical qubit subsets. That is, by utilising more than two time periods for the logical qubit operation, the length of the logical chain of synchronisation groups can be increased, while minimising the synchronisation requirements of the quantum computing system.

[0069] A further example implementation is shown in Figure 4. The example of Figure 4 includes the same arrangement of physical qubit subsets 320 and logical qubits 310 as discussed in relation to Figure 3. However, in the example of Figure 4, the logical qubit operation is executed over only two time periods 330(1)-(2), thus minimising the time over which the logical qubit operation is executed, whilst reducing the synchronisation requirements (i.e. the amount of qubits that should be synchronised with one another at any given time) for the logical qubit operation.

[0070] In the first time period 330(1), two initial synchronisation groups 340(1)a, 340(1)b are defined. The first initial synchronisation group 340(1)a comprises physical qubit subsets 320(1)- (3), and the second initial synchronisation group 340(1)b comprises physical qubit subsets 320(5)-(7). That is, the physical qubits subsets 320(1)-(3) are synchronised with one another during the first time period 330(1), and the physical qubits subsets 320(5)-(7) are synchronised with one another during the first time period 330(1). However, the first and second initial synchronisation groups 340(1)a, 340(1)b are not required to be synchronised with one another. This is shown by solid vertical lines separating the first and second initial synchronisation groups 340(1)a, 340(1)b. During the second time period 330(2), a single later synchronisation group 340(2) is defined, comprising physical qubit subsets 320(3)-(5). The later synchronisation group 340(2) partially overlaps both the first and second initial synchronisation groups 340(1)a, 340(1)b, such that at least one physical qubit subset included in the first and second initial synchronisation groups 340(1)a, 340(1) is also included in the later synchronisation group 340(2).

[0071] Therefore, as can be seen in Figure 4, the synchronisation groups 340(1)a, 340(2), 340(1)b form a logical chain of partially overlapping synchronisation groups between the first physical qubit subset 320(1) and the second physical qubit subset 320(7), and spanning two time periods 330(1)-(2), where two separate (non-overlapping) synchronisation groups are present in the first time period 330(1). As such, the overall time taken to execute the logical qubit operation can be minimised while also keeping the synchronisation requirements limited.

[0072] The approach discussed in relation to Figure 4, whereby multiple separate synchronisation groups are defined for a single time period, is applicable in a more general sense, such that multiple synchronisation groups may be included in various time periods, in any combination. One such additional example is shown in Figure 5, where identical reference numerals denote entities discussed in relation to Figures 3 and 4 using the same reference numerals, except where otherwise stated. In the example of Figure 5, a single initialsynchronisation group 340(1) is defined for the first time period 330(1), while two later synchronisation groups 340(2)1 , 340(2)b are defined for the second time period 330(2). More specifically, the initial synchronisation group 340(1) comprises physical qubit subsets 320(3)-(5), the first later synchronisation group 340(2)a comprises physical qubit subsets 320(1 )-(3), and the second later synchronisation group 340(2)b comprises physical qubit subsets 320(5)-(7).

[0073] Furthermore, as can be seen in Figure 5, the first and second physical qubit subsets 320(1), 320(7) do not form respective synchronisation groups with other physical qubit subsets during the first time period (as in Figure 4), but rather form part of synchronisation groups with other physical qubit subsets in a later (in this specific case, the second) time period. This is generally true, such that the first and second physical qubit subsets 320(1), 320(7) may be part of a synchronisation group for substantially any time period for the logical qubit operation. Therefore, during the first time period, the physical qubits of the first physical qubit subset are synchronised only with one another, and as such the first physical qubit subset may be considered to form its own synchronisation group including only the first physical qubit subset, or alternatively the first physical qubit subset may be considered to not be part of a synchronisation group during the first time period. Similarly, during the first time period, the physical qubits of the second physical qubit subset are synchronised only with one another, and as such the second physical qubit subset may be considered to form its own synchronisation group including only the second physical qubit subset, or alternatively the second physical qubit subset may be considered to not be part of a synchronisation group during the first time period.

[0074] Moreover, while the examples of Figures 4 and 5 illustrate the first and second physical qubit subsets 320(1), 320(7) which implement the first and second logical qubits 310(1)-(2) as being part of separate synchronisation groups which are defined for the same time period, it should be appreciated that the synchronisation groups comprising the first and second physical qubit subsets may be defined for different time periods of the logical qubit operation. This example is illustrated in Figure 6. In the example of Figure 6, a logical qubit operation between a first logical qubit 410(1) and a second logical qubit 410(2) is performed. The first logical qubit 410(1) is implemented by a first physical qubit subset 420(1) and the second logical qubit 410(2) is implemented by a second physical qubit subset 420(9). The quantum computing system on which the logical qubit operation is performed also includes additional physical qubits, including seven additional physical qubit subsets 420(2)-(8). In the present example, the first and second physical qubit subsets 420(1), (9) each have a size d, and the quantum computing system includes additional physical qubit subsets 420(2)-(8), where each of the additional physical qubit subsets 420(2)-(8) has a size of d. While in this example each of the physical qubit subsets 420(1)-(9) are shown as having a size d, it should be appreciated that the physical qubit subsets may have different sizes, as discussed in relation to Figures 2 and 3.

[0075] In a first time period 430(1) between times ti and t2, two initial synchronisation groups 440(1) are defined. A first initial synchronisation group 440(1)a comprises physical qubit subsets 420(1)-(3), and a second initial synchronisation group 440(1)b comprises physical qubit subsets 420(5)-(7). In a second time period 430(2) between times t2and t3, two later synchronisation groups 440(2) are defined. A first later synchronisation group 440(2)a comprises physical qubit subsets 420(3)-(5), and a second later synchronisation group 440(2)b comprises physical qubit subsets 420(7)-(9). As such, the first physical qubit subset 420(1) and second physical qubit subset 420(9) are included in synchronisation groups which are defined for different time periods. Accordingly, a logical chain of partially overlapping synchronisation groups can be defined in substantially any manner in order to minimise the number of physical qubits which are to be synchronised at any given time.

[0076] As such, logical chains of partially overlapping synchronisation groups can be defined including substantially any number of physical qubits, divided into any number of synchronisation groups. Accordingly, substantially any logical qubit operation can be performed in only two time periods, while utilising any number of additional physical qubits (other than the physical qubit utilised for implementing the logical qubits). A further example of this is shown in Figure 7. In the example of Figure 7, a logical qubit operation between a first logical qubit 510(1) and a second logical qubit 510(2) is performed. The first logical qubit 510(1) is implemented by a first physical qubit subset 520(1) and the second logical qubit 510(2) is implemented by a second physical qubit subset 520(11). The quantum computing system on which the logical qubit operation is performed also includes additional physical qubits, including nine additional physical qubit subsets 520(2)-(10). In the present example, the first and second physical qubit subsets 520(1), (11) each have a size d. The quantum computing system includes additional physical qubit subsets 520(2)-(10), where each of the additional physical qubit subsets 520(2)-(10) has a size of d. While in this example each of the physical qubit subsets 520(1)-(11) are shown as having a size d, it should be appreciated that the physical qubit subsets may have different sizes, as discussed in relation to Figures 2, 3, and 6.

[0077] In a first time period 530(1) between times t1 and t2, three initial synchronisation groups 540(1) are defined. A first initial synchronisation group 540(1)a comprises physical qubit subsets 520(1)-(3), a second initial synchronisation group 540(1)b comprises physical qubit subsets 520(5)-(7), and a third initial synchronisation group 540(1)c comprises physical qubit subsets 520(9)-(11). In a second time period 430(2) between times t2and t3, two later synchronisation groups 540(2) are defined. A first later synchronisation group 540(2)a comprises physical qubit subsets 520(3)-(5), and a second later synchronisation group 540(2)b comprises physical qubit subsets 420(9)-(11). Accordingly, a logical chain of five partially overlapping synchronisation groups spanning two time periods 530(1)-(2) is defined between the first and second physical qubit subsets 520(1), (11), and including nine additional physical qubit subsets 520(2)-(10). Therefore, as demonstrated by the example of Figure 7, arbitrarily long logical chains of partially overlapping synchronisation groups can be defined, so as to perform logical qubit operations including arbitrarily many numbers of additional physical qubits.

[0078] The approaches used in Figures 6 and 7 (which have multiple distinct synchronisation groups in each of the initial and later time periods) can be applied to physical qubit subsets with arbitrary separations whilst only requiring two time periods. In other words, the approaches used in Figures 4-7 can be used to perform logical operations with fixed-size synchronisation groups in just two time periods regardless of the separation between the logical qubits. This is particularly advantageous because it means that the number of physical qubits that need to be synchronised (i.e. the size of the synchronisation groups) and the time required to perform the operation are both independent of the separation between the logical qubits (i.e. the size of the synchronisation groups does not scale with the separation (unlike the prior art approach in Figure 1), and the time required to perform the logical operation does not scale with the separation (unlike the approach in Figure 3)).

[0079] Accordingly, in performing the logical qubit operation over only two time periods (i.e. with synchronisation groups having multiple physical qubit subsets defined only for two time periods), the time taken to perform a given logical qubit operation can be essentially constant, regardless of the number of additional physical qubits (otherthan the physical qubits utilised for implementing the logical qubits themselves) which are to be used for the logical qubit operation. That is, the number of physical qubits which are collectively synchronised in any given logical qubit operation may be independent of the operation itself. As such, additional techniques to maintainsynchronisation between large sets of physical qubits, such as re-synchronisation during the operation and / or artificially delaying signals, can be minimised, not only reducing the required complexity of the control systems for quantum computing systems, but keeping the time taken for a particular type of logical qubit operation constant. This may be particularly advantageous for logical qubit operations performed using large numbers of physical qubits.

[0080] The aforementioned examples illustrated in Figures 2-7 relate primarily to cases where the logical qubit operation comprises an entangling operation between the two logical qubits. However, it should be appreciated that the techniques of the present disclosure are applicable to other types of logical qubit operation, for example a relocation operation of one or more logical qubits. It should be understood that the relocation of a logical qubit refers to the relocation of the encoded state of a logical qubit to different physical qubits (e.g. from one physical qubit subset to another), with the physical qubits themselves having fixed positions. Moreover, it should also be understood that in some cases the logical qubit operation may include both a relocation of a logical qubit, and an entangling operation between multiple logical qubits.

[0081] An example implementation is shown in Figure 8. In the example of Figure 8, a logical qubit operation is performed in which the state of a first logical qubit 610(1) is relocated to different physical qubits, so as to form a second logical qubit 610(2). The first logical qubit 610(1) is (initially) implemented by a first physical qubit subset 620(1). The quantum computing system on which the logical qubit operation is performed also includes a second physical qubit subset 620(5) to which the state of the first logical qubit 610(1) is to be relocated to form the second logical qubit 610(2). The quantum computing system on which the logical qubit operation is performed also includes additional physical qubits, including three additional physical qubit subsets 420(2)-(4). In the present example, the first and second physical qubit subsets 620(1), (5) each have a size d. The quantum computing system includes additional physical qubit subsets 620(2)-(4), where each of the additional physical qubit subsets 420(2)-(4) has a size of d. While in this example each of the physical qubit subsets 620(1)-(5) are shown as having a size d, it should be appreciated that the physical qubit subsets may have different sizes, as discussed in relation to Figures 2, 3, 6 and 7.

[0082] In a first time period 630(1) between times ti and t2, an initial synchronisation group 540(1) is defined comprising physical qubit subsets 620(1)-(3). In a second time period 630(2) between times t2and t3, a later synchronisation group is defined comprising physical qubit subsets 620(3)- (5). As can also be seen in Figure 8, after time ti, the physical qubits of the first physical qubit subset 620(1) are no longer required to be synchronised with another, and instead from time t2onwards, physical qubits of the second physical qubit subset 620(5) are to be synchronised with one another in order to implement the second logical qubit 610(2). In this manner, the encoded state corresponding to the first logical qubit 610(1) is moved from the first physical qubit subset 620(1) to the second physical qubit subset 620(5). As such, the logical qubit is relocated to different physical qubits. After time t3, the physical qubits of the second physical qubit subset 620(5) remain synchronised with one another in order for the state of the second logical qubit 610(2) to be persisted. Therefore, as can be seen, logical qubit operations including a relocation operation can be performed utilising synchronisation schemes spanning multiple time periods, according to the present disclosure. It should be appreciated that the features discussed in relation to Figure 8 are compatible with each of the example teachings of the present disclosure discussed herein.

[0083] In each of the aforementioned examples illustrated in Figures 2-8, the time periods for which synchronisation groups are defined are contiguous. However, it should be appreciated thatthe aforementioned examples are compatible with cases where the time periods for which synchronisation groups are defined are not contiguous, and one or more intermediate time periods may be present in the logical qubit operation in which a particular physical qubit subset is not required to be synchronised with any other physical qubit subsets. This is illustrated in Figure 9.

[0084] In the example of Figure 9, a logical qubit operation between a first logical qubit 710(1) and a second logical qubit 710(2) is performed. The first logical qubit 710(1) is implemented by a first physical qubit subset 720(1) and the second logical qubit 710(2) is implemented by a second physical qubit subset 720(5). The quantum computing system on which the logical qubit operation is performed also includes additional physical qubits, including three additional physical qubit subsets 720(2)-(4). In the present example, the first and second physical qubit subsets 720(1), (5) each have a size d. The quantum computing system includes additional physical qubit subsets 720(2)-(4), where each of the additional physical qubit subsets 720(2)-(4) has a size of d. While in this example each of the physical qubit subsets 720(1)-(5) are shown as having a size d, it should be appreciated that the physical qubit subsets may have different sizes, as discussed in relation to Figures 2, 3, 6-8. In a first time period 730(1) between times ti and t2, an initial synchronisation group 740(1) is defined comprising physical qubit subsets 720(1)-(3). In a second time period, no synchronisation groups are defined such that there is no requirement for synchronisation between multiple physical qubit subsets. That is, the physical qubits of physical qubit subset 720(3) are to remain synchronised with one another, but the physical qubit subset 720(3) is not synchronised with any other physical qubit subsets during the second time period 730(2). In a third time period 730(3) between times t3and t4, a later synchronisation group is defined comprising physical qubit subsets 720(3)-(5).

[0085] In this way, the time periods for which synchronisation groups are defined for the logical qubit operation need not be contiguous. As such, the manner in which the synchronisation scheme is defined, and hence the manner in which the logical qubit operation is executed, is flexible, and may therefore be adapted based on various factors, such as the capabilities of the quantum computing system and / or concurrent operations being executed by the quantum computing system. It should be appreciated that the features discussed in relation to Figure 9 are compatible with each of the example teachings of the present disclosure discussed herein.

[0086] In the previous examples, each of the synchronisation groups includes the same number of physical qubit subsets. In other words, each of the synchronisation groups has the same code distance. However, this is not necessarily required, as the various synchronisation groups may have different code distances / sizes.

[0087] In order to further explain the manner in which the techniques according to the present disclosure may be implemented, Figures 10A-D illustrate the synchronisation between physical qubits of the quantum computing system on a surface code during a logical qubit operation, for the example synchronisation scheme shown in Figure 4. In the present examples, the logical qubit operation is a lattice surgery entangling operation comprising a logical ZZ measurement. Figure 10A is a view of a surface code representing physical qubits at time to, before the logical qubit operation has begun. While it is often preferable that the physical qubits are arranged in a regular array, this is not essential to perform quantum computation using the planar code, and the qubits could alternatively have a different physical arrangement while still achieving the desired connectivity (e.g. degree- four connectivity or greater).

[0088] As can be seen, the physical qubits include physical qubits subsets 320(1)-(7). The first physical qubit subset 320(1) implements the first logical qubit 310(1), and the second physicalqubit subset 320(5) implements the second logical qubit 310(2), where each physical qubit subset may include both data qubits shown as black dots, and auxiliary (i.e. syndrome) qubits shown as white dots. As can be seen, some data qubits may be considered to belong to two physical qubit subsets, while other data qubits may be considered to belong only one physical qubit subset. That is, the physical qubit subsets 320(1)-(7) may be considered to partially overlap one another, such that some data qubits of a given physical qubit subset are shared with another physical qubit subset. The physical qubits subsets have a code distance of three (i.e. an undetectable logical qubit error requires at least three physical errors, and the code is capable of reliably detecting and correcting any combination of up to [code distance / 2J = [3 / 2J = 1. errors) and 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. It should be understood that the code distance is a global property of the quantum error correction (QEC) code and is well understood by the person skilled in the art.

[0089] The surface codes for the two logical qubits 310(1)-(2) are formed of Z-type stabilisers and X-type stabilisers, as shown in Figure 10A. Z-type stabilisers involve joint Pauli-Z measurements (e.g. Z Z Z Z measurements) of the data qubits 202 at the vertices of the Z-type stabilisers 302 (four data qubits for the square stabilisers and only two data qubits for the triangular stabilisers because the third vertex is occupied by a syndrome qubit). Similarly, the X- type stabilisers involve joint Pauli-X measurements of the data qubits at the vertices of the X-type stabilisers. Each stabiliser is associated with a single syndrome qubit (either in the centre of the respective square, or at a vertex of the respective triangle). It can be seen that the syndrome qubit of each stabiliser is connected to the data qubits associated with that stabiliser. All of the stabilisers commute with each other.

[0090] The syndrome qubits are used to measure the value of the stabilisers, e.g. using CNOT or CZ gates between the syndrome qubit and its neighbouring data qubits, for the X-type stabilisers and Z-type stabilisers respectively. The stabilisers detect the ends of chains of errors on the data qubits. The Z-type stabilisers detect the ends of chains of errors involving X and / or Y data qubit errors, and the X-type stabilisers detect the ends of chains of errors involving Y and / or Z data qubit errors. Moreover, it should be appreciated that detection and correction of Z (and Y) data qubit errors can be achieved using equivalent procedures to those used for detecting and correcting X errors by Z-type stabilisers.

[0091] Returning to Figure 10A, between times to and ti, syndrome extraction is performed on the first and second physical qubit subsets 320(1), 320(7) (i.e. for the first and second logical qubits 320(1), 320(2)), using techniques known in the art. Syndrome extraction therefore includes the application of physical gates, otherwise referred to as physical qubit operations. This may be done using control hardware for the quantum computing system in which the physical qubits are located, for example using laser pulses. The timing of the physical qubit operations for the syndrome extraction for the logical qubits 310(1)-(2) is synchronised (to within a predetermined margin of error). In this way, certain physical qubit operations on the first and second physical qubit subsets 320(1), 320(7) may be performed substantially simultaneously, such that multiple qubit operations may be performed simultaneously in a single layer of an operation. Furthermore, before time ti, the qubits (i.e. physical qubit subsets) included in synchronisation group(s) for the first time period (beginning at time ti) may be synchronised (i.e. reset in the X basis), such that the included in synchronisation group(s) for the first time period are synchronised for the start of the first time period.

[0092] Figure 10B shows the same example during the second time period from times ti to t2, as illustrated in Figure 4. During this time period, syndrome extraction is performed for each of the two initial synchronisation group(s) 340(1)a, 340(1)b. For example, a product of stabiliser measurements may be taken from the syndrome qubits of Z stabilisers included in the physical qubit subsets 320(2)-(3) and the physical qubit subsets 320(5)-(6), to form part of the logical ZZ measurement. Each of these stabiliser measurements may be taken from any round of syndrome extraction. The physical qubits subsets of the first initial synchronisation group 340(1)a are synchronised with one another such that physical qubit operations on the physical qubits of the first initial synchronisation group 340(1)a begin substantially simultaneously (i.e. to within a predetermined margin of error). Similarly, the physical qubits subsets of the second initial synchronisation group 340(1)b are synchronised with one another such that physical qubit operations on the physical qubits of the second initial synchronisation group 340(1)b begin substantially simultaneously (i.e. to within a predetermined margin of error). In general, various techniques known in the art may be used in combination with the example teachings of the present disclosure in order to synchronise qubits and / or the control hardware for said qubits according to the synchronisation schemes discussed herein.

[0093] After syndrome extraction has completed for the initial synchronisation group(s) 340(1)a, 340(1)b, the data qubits belonging only to physical qubit subsets 320(2), 320(6) are measured. That is, as can be seen from Figure 10B, physical qubit subset 320(2) includes nine data qubits, where three of these data qubits are shared with physical qubit subset 320(1), three of the data qubits are shared with physical qubit subset 320(3), and three of the data qubits belong only to physical qubit subset 320(2). Similarly, physical qubit subset 320(6) includes nine data qubits, where three of these data qubits are shared with physical qubit subset 320(5), three of the data qubits are shared with physical qubit subset 320(7), and three of the data qubits belong only to physical qubit subset 320(6). The data qubits belonging only to physical qubit subsets 320(2), 320(6) are therefore measured. In this way, the data qubits which are not included in a synchronisation group (or logical qubit) for the upcoming time period are measured. Furthermore, before time t2, the physical qubits of physical qubit subset 320(4) which belong only to physical qubit subset 320(4) (i.e. the three centre-most data qubits) are reset in the X basis. While this reset of the data qubits may occur earlier during the logical qubit operation, the later this reset is performed (prior to time t2) the less idle noise is generated on those qubits.

[0094] The process then proceeds to the second time period, from time t2to t3, as shown in Figure 10C. Syndrome extraction is performed on the later synchronisation group 340(2) and the logical qubits 310(1 )-(2). A product of stabiliser measurements may be taken from the syndrome qubits of Z stabilisers included physical qubit subset 320(4), and multiplied to contribute to the logical ZZ measurement outcome. Each of these stabiliser measurements may be taken from any round of syndrome extraction. Furthermore, for stabilisers present in both the second time period (shown in Figure 10C) and the first time period (shown in Figure 10B) of the logical qubit operation, a stabiliser measurement can be taken in either the first or second time periods. After syndrome extraction has completed for the later synchronisation group 340(2), the data qubits included in the later synchronisation group 340(2) are measured in the X basis. As such, a ZZ logical measurement operation has been performed between the logical qubits 320(1), 320(2).

[0095] Figure 10D illustrates the same example after time t3, after the logical qubit operation has concluded. Each of the first and second physical qubit subsets 320(1), 320(7) remain synchronised with one another, however, as discussed above, a ZZ logical measurement operation has been performed between the logical qubits 320(1), 320(2).

[0096] While Figures 10A-10D illustrate a ZZ logical measurement operation (which is an example of a lattice surgery operation that entangles the two logical qubits), it should be appreciated that the techniques described in relation to Figures 10A-10D are generally applicable to other types of logical qubit operation. Moreover, the techniques described in relation to Figures 10A-D are also more generally applicable to other example arrangements other than that of Figure 4. Furthermore, it should be appreciated that the specific operations described in relation to Figures 10A-10D are only one example implementation and that other techniques for executing the logical qubit operation may instead be implemented in combination with the synchronisation schemes discussed in the present disclosure.

[0097] Furthermore, while the present disclosure generally discusses logical qubit operations between two logical qubits, it should be appreciated that the techniques of the present disclosure are more generally applicable to logical qubit operations between two or more logical qubits. For example, three or more logical qubits may be simultaneously entangled with one another in a single logical qubit operation. That is, instead of entangling a first logical qubit with a second logical qubit, and then entangling the second logical qubit with a third logical qubit, the logical qubit operation may entangle each of the first, second and third logical qubits with one another at the same time, using synchronisation schemes as discussed herein.

[0098] The methods discussed herein may be implemented in various systems, such as the system 900 shown in Figure 11. As shown in Figure 11 , a logical layer 920 receives one or more quantum algorithms 910 to be executed by a quantum computing system 930, and may be configured to convert the quantum algorithm 910 from a logical circuit to a physical quantum circuit to be executed by a specific quantum computing system. For example, the logical layer 920 may be configured to first convert 921 the logical circuit to FT primitives, and then convert 922 the FT primitives to physical quantum circuit. However, it should be appreciated that other approaches for converting the quantum algorithm 910 from a logical circuit to a physical quantum circuit may be utilised.

[0099] The logical layer 920 may be implemented by a computing system, such as computing system 1000 illustrated in Figure 12. The computing system may include one or more processors 1010, one or more memories 1020, a network interface 1030, one or more input / output (I / O) components 1040, and / or data storage 1050. The components of the computing system 1000 may be coupled to one another to provide the functionality of the logical layer 920. The computing system 1000 may form part of the quantum computing system 930, and / or quantum error correction system 940 shown in Figure 11 , or may be implemented by a standalone computing system, separate from the quantum computing system 930 and quantum error correction system 940 shown in Figure 11. The logical layer 920 may receive the quantum algorithm(s) 910 via a user interface (such as a graphical user interface (GUI)) using one or more input / output (I / O) components 1040, or an indication of the quantum algorithm(s) 910 may be received from a further system via one or more communication components (such as a network interface 1030).

[0100] The logical layer 920 may determine the physical circuits based on an indication of the QPU architecture 953 for the quantum computing system 953. That is, the logical layer 920 may receive an indication of the QPU architecture 953, e.g. from the quantum computing system 930 via a network interface 1030, or directly via an I / O components 1040. This indication of the QPU architecture 953 may e.g. include an indication of the native gates for the quantum computing system 930 (i.e. the physical quantum gates which can be implemented by the quantum computing system 930). Accordingly, the logical layer 920 may compile the quantum algorithm 910 into a combination of said native gates. Furthermore, the indication of the QPU architecture953 may indicate the synchronisation capabilities of the quantum computing system 930 (e.g. the synchronisation capabilities of the control system 932 of the quantum computing system 930), such as a number of physical qubits which the control system 932 is capable of synchronising for a given time period.

[0101] In addition, the logical layer 920 may determine a synchronisation scheme according to which quantum algorithm 910, or the physical quantum circuit, is to be executed. The synchronisation scheme may be determined based on a variety of factors, such as a number of additional physical qubits (i.e. a number of additional physical qubit subsets) to be utilised for the logical qubit operation, the synchronisation capabilities of the quantum computing system 930, a number of logical qubits included in the logical qubit operation, and / or one or more other factors. The logical layer 920 may then provide (e.g. transmit) an indication of the synchronisation scheme to the quantum computing system 930 and / or the QEC system 940. As mentioned above, in some cases the logical layer 920 may be implemented as part of the quantum computing system 930 and / or QEC system 940, and as such the provision of the indication of the synchronisation scheme may not necessarily include transmission of data between different physical entities.

[0102] Furthermore, in some cases, the functionality of the logical layer 920 may be distributed such that the determination of the physical quantum circuit and the determination of the synchronisation scheme may be performed by different entities. For example, the determination of the physical quantum circuit based on the quantum algorithm 910 may be performed at a separate computing system, while the determination of the synchronisation scheme may be performed at quantum computing system 930 and / or the QEC system 940. That is, in some cases the logical layer may provide an indication of the physical quantum circuit to the quantum computing system 930 and / or the QEC system 940, where the physical quantum circuit is indicative of the synchronisation system to be used, such that the quantum computing system 930 and / or the QEC system 940 may determine the synchronisation scheme based on the indication of the physical quantum circuit.

[0103] The logical layer 920 provides an indication 951 , 952 of the quantum circuit to the quantum computing system and / or the QEC system 940. The quantum computing system 930 includes physical qubits 938 (or other quantum devices, such as qutrits or qudits) and a control system 932 for manipulating the state of the qubits 938. The quantum computing system 930 may additionally include transpilation components configured to optimise and / or compile the quantum circuit, however this process may in some cases be carried out by the logical layer 920 (although the logical layer 920 as a whole may be implemented at the quantum computing system 930 in some cases). The control system 932 receives a final physical quantum circuit and executes this quantum circuit on the qubits 938. As part of this process, the control system 932 may generate sequence instructions 933 for controlling the control hardware, for example to generate laser pulses 934 to manipulate the state of qubits 938 and / or perform multi-qubit gates. In addition, the control system 932 may include hardware and / or software components for capturing 935 syndrome data 935 and / or readout data from the qubits 938. Furthermore, in some cases the control system 932 may be configured to discriminate 936 readout data, e.g. by altering the format of the readout data, or discarding unneeded / unwanted readout data.

[0104] The control system 932 may additionally include a QEC interface 937 to allow the quantum computing system 930 to communicate with the QEC system 940. The QEC system 940 may also include its own QEC interface 943 to allow the QEC system 940 to communicate with the quantum computing system 930. The quantum computing system 930 may transmit syndrome data 954 to the QEC system 940. The syndrome data 954 may in some cases be in the form ofreadout data, including physical qubit measurement values. The QEC system 940 may process the syndrome data 954 to detect errors in the state of qubits 938 of the quantum computing system 930. For example, the QEC system 940 may in some cases perform routing and pre-processing of the syndrome data 954 to organise the syndrome data 954 (and in some cases discard unwanted syndrome data) into a format to allow decoding of the syndrome data 954. The QEC system 940 may additionally be configured to decode the syndrome data 954 and provide an output indicative of errors. Moreover, the QEC system 940 may identify corrections to be made to the syndrome data 954 and / or state of the qubits 938. This may include corrections to the state of one or more physical qubits, and / or one or more logical qubits. In some cases, the QEC system 940 may transmit an indication 955 of the identified errors and / or corrections to the quantum computing system 930. The control system 932 of the quantum computing system 930 may use the indication 955 of the identified errors and / or corrections to manipulate the state of one or more qubits 938 to perform one or more corrections.

[0105] Accordingly, the example system 900 shown in Figure 11 may implement the techniques according to the teachings of the present disclosure. For example, the quantum computing system 930 may receive, from the logical layer 920, an indication of: a logical qubit operation between a first logical qubit and a second logical qubit to be executed by the quantum computing system 930, and a synchronisation scheme, wherein the first logical qubit is implemented by a first physical qubit subset of the quantum computing system, a second logical qubit is implemented by a second physical qubit subset of the quantum computing system, and wherein the plurality of physical qubits of the quantum computing system further includes one or more additional physical qubit subsets, wherein the synchronisation scheme defines a plurality of time periods of the logical qubit operation and one or more synchronisation groups for each of the plurality of time periods, wherein each of the synchronisation groups includes two or more physical qubit subsets to be synchronised with one another for a respective time period, and wherein the synchronisation groups partially overlap with one another to form a logical chain of synchronisation groups between the first and second physical qubit subsets spanning the plurality of time periods. The quantum computing system 930 may then execute the logical qubit operation by: in an initial time period, performing one or more physical qubit operations on physical qubits within one or more initial synchronisation groups for the initial time period defined by the synchronisation scheme; and in a later time period subsequent to the first time period, performing one or more physical qubit operations on physical qubits within one or more subsequent synchronisation groups for the later time period defined by the synchronisation scheme.

[0106] Furthermore, the logical layer 920 may identify a logical qubit operation to be performed on a plurality of logical qubits and the quantum computing system 930 on which the logical qubit operation is to be performed. The logical layer 920 may additionally, based on the identified logical qubit operation and quantum computing system 930, identify the synchronisation scheme for the logical qubit operation. The logical layer 920 may additionally transmit an indication of the logical qubit operation and the synchronisation scheme to the quantum computing system 930.

[0107] In addition, the quantum error correction system 940 may receive, from the logical layer 920, an indication of a logical qubit operation to be performed on the quantum computing system 930 and the aforementioned synchronisation scheme. The QEC system 940 may additionally receive syndrome data from the quantum computing system 930 for the logical qubit operation, wherein receiving syndrome data comprises: in an initial time period, receiving initial syndrome data for physical qubits within one or more initial synchronisation groups for the initial time period defined by the synchronisation scheme; and in a later time period subsequent to the first timeperiod, receiving subsequent syndrome data for physical qubits within one or more subsequent synchronisation groups for the later time period defined by the synchronisation scheme.

[0108] It should be noted that the system 900 shown in Figure 11 is just one example system, and that the techniques of the present disclosure may implement one or more other systems different to that discussed in relation to Figure 11.

[0109] Furthermore, the methods described herein may also be embodied or encoded in a computer-readable medium, such as a computer-readable storage medium, containing instructions. Instructions embedded or encoded in a computer-readable medium may cause a programmable processor, or other processor, to perform the method, e.g., when the instructions are executed. Computer-readable media may include non-transitory computer-readable storage media and transient communication media. Computer readable storage media, which is tangible and non-transitory, may include random access memory (RAM), read only memory (ROM), programmable read only memory (PROM), erasable programmable read only memory (EPROM), electronically erasable programmable read only memory (EEPROM), flash memory, a hard disk, a CD-ROM, a floppy disk, a cassette, magnetic media, optical media, or other computer-readable storage media. The term “computer-readable storage media” refers to physical storage media, and not signals, carrier waves, or other transient media. As noted above, computer readable media may include transient communication media. Such communication media may occur within a single computer system or between multiple computer systems, and may take the form of transient signal-conveying media such as carrier waves and transmission signals.

[0110] Figure 13 illustrates a method 1300 for maintaining synchronisation between data qubits during a logical qubit operation between a plurality of logical qubits implemented by a quantum computing system, according to an example of the present disclosure, where the quantum computing system includes a plurality of physical qubits implementing the plurality of logical qubits, wherein the plurality of physical qubits includes a plurality of physical qubit subsets. The method 1300 comprises step 1310 of receiving an indication of: a logical qubit operation between a first logical qubit and a second logical qubit to be executed by the quantum computing system, and a synchronisation scheme, wherein the first logical qubit is implemented by a first physical qubit subset of the quantum computing system, a second logical qubit is implemented by a second physical qubit subset of the quantum computing system, and wherein the plurality of physical qubits of the quantum computing system further includes one or more additional physical qubit subsets, wherein the synchronisation scheme defines a plurality of time periods of the logical qubit operation and one or more synchronisation groups for each of the plurality of time periods, wherein each of the synchronisation groups includes two or more physical qubit subsets to be synchronised with one another for a respective time period, and wherein the synchronisation groups partially overlap with one another to form a logical chain of synchronisation groups between the first and second physical qubit subsets spanning the plurality of time periods.

[0111] The method 1300 then includes step 1320 of executing the logical qubit operation. This step 1320 includes sub-step 1321 of: in an initial time period, performing one or more physical qubit operations on physical qubits within one or more initial synchronisation groups for the initial time period defined by the synchronisation scheme. Step 1320 also includes sub-step 1322 of: in a later time period subsequent to the first time period, performing one or more physical qubit operations on physical qubits within one or more subsequent synchronisation groups for the later time period defined by the synchronisation scheme.

[0112] Figure 14 illustrates a method 1400 comprising step 1410 of identifying a logical qubit operation to be performed on a plurality of logical qubits and a quantum computing system onwhich the logical qubit operation is to be performed. The method also includes step 1420 of, based on the identified logical qubit operation and quantum computing system, identifying a synchronisation scheme for the logical qubit operation, wherein the synchronisation scheme defines a plurality of time periods of the logical qubit operation and synchronisation groups for each of the plurality of time periods, wherein each of the synchronisation groups includes two or more physical qubit subsets of the quantum computing system to be synchronised with one another for a respective time period, and wherein the synchronisation groups partially overlap with one another to form a logical chain of synchronisation groups between physical qubit subsets implementing the plurality of logical qubits, wherein the logical chain of synchronisation groups spans the plurality of time periods. The method 1400 further includes step 1430 of transmitting an indication of the logical qubit operation and the synchronisation scheme to the quantum computing system.

[0113] Figure 15 illustrates a method 1500 comprising step 1510 of: receiving an indication of a logical qubit operation to be performed on a quantum computing system and a synchronisation scheme, wherein the synchronisation scheme defines a plurality of time periods of the logical qubit operation for a plurality of logical qubits implemented by the quantum computing system and synchronisation groups for each of the plurality of time periods, wherein each of the synchronisation groups includes two or more physical qubit subsets of the quantum computing system to be synchronised with one another for a respective time period, and wherein the synchronisation groups partially overlap with one another to form a logical chain of synchronisation groups between physical qubit subsets implementing the plurality of logical qubits, wherein the logical chain of synchronisation groups spans the plurality of time periods.

[0114] The method additionally includes step 1520 of receiving syndrome data from the quantum computing system for the logical qubit operation. Step 1520 includes sub-step 1521 of: in an initial time period, receiving initial syndrome data for physical qubits within one or more initial synchronisation groups for the initial time period defined by the synchronisation scheme. Step 1520 additionally includes step 1522 of: in a later time period subsequent to the first time period, receiving subsequent syndrome data for physical qubits within one or more subsequent synchronisation groups for the later time period defined by the synchronisation scheme.As such from one perspective, there has been provided methods, quantum computing systems, systems, and computer-readable media are provided for synchronising qubits during a logical qubit operation. The logical qubit operation is executed according to a synchronisation scheme which defines synchronisation groups comprising partially overlapping sets of qubits for different time periods of the logical qubit operation, where the synchronisation groups form a logical chain of partially overlapping synchronisation groups between two logical qubits, to reduce the number of physical qubits which must be synchronised with one another.

Claims

CLAIMS1 . A method for maintaining synchronisation between data qubits during a logical qubit operation between a plurality of logical qubits implemented by a quantum computing system, wherein the quantum computing system includes a plurality of physical qubits implementing the plurality of logical qubits, wherein the plurality of physical qubits includes a plurality of physical qubit subsets, wherein the method comprises: receiving an indication of: a logical qubit operation between a first logical qubit of the plurality of logical qubits and a second logical qubit of the plurality of logical qubits to be executed by the quantum computing system, and a synchronisation scheme, wherein the first logical qubit is implemented by a first physical qubit subset of the quantum computing system, a second logical qubit is implemented by a second physical qubit subset of the quantum computing system, and wherein the plurality of physical qubits of the quantum computing system further includes one or more additional physical qubit subsets, wherein the synchronisation scheme defines a plurality of time periods of the logical qubit operation and one or more synchronisation groups for each of the plurality of time periods, wherein each of the synchronisation groups includes two or more physical qubit subsets to be synchronised with one another for a respective time period, and wherein the synchronisation groups partially overlap with one another to form a logical chain of synchronisation groups between the first and second physical qubit subsets spanning the plurality of time periods; and executing the logical qubit operation by: in an initial time period, performing one or more physical qubit operations on physical qubits within one or more initial synchronisation groups for the initial time period defined by the synchronisation scheme; and in a later time period subsequent to the first time period, performing one or more physical qubit operations on physical qubits within one or more subsequent synchronisation groups for the later time period defined by the synchronisation scheme.

2. The method according to claim 1 , wherein the one or more initial synchronisation groups includes the first qubit subset, and wherein the one or more later synchronisation groups include the second qubit subset.

3. The method according to claim 1 , wherein the one or more initial synchronisation groups include: a first initial synchronisation group comprising the first qubit subset, and one or more other second initial synchronisation groups comprising additional physical qubit subsets, wherein the one or more later synchronisation groups include: a first later synchronisation group comprising the second qubit subset, and one or more other later synchronisation groups, wherein the one or more other second initial synchronisation groups partially overlap with the first later synchronisation group and the one or more other later synchronisation groups, and wherein the one or more other later synchronisation groups partially overlap with the first initial synchronisation group and the one or more other initial synchronisation groups.

4. The method according to claim 1 , wherein the one or more initial synchronisation groups includes a first initial synchronisation group and a second initial synchronisation group, wherein the first initial synchronisation group includes the first qubit subset, and wherein the second initial synchronisation group includes the second qubit subset, and wherein the one or more later synchronisation groups partially overlap with the first initial synchronisation group and the second initial synchronisation group.

5. The method according to claim 1 , wherein the one or more later synchronisation groups includes a first later synchronisation group and a second later synchronisation group, wherein the first later synchronisation group includes the first qubit subset, and wherein the second later synchronisation group includes the second qubit subset, and wherein the one or more initial synchronisation groups partially overlap with the first later synchronisation group and the second later synchronisation group.

6. The method according to any preceding claim, wherein the method further comprises: in a further time period between the initial and later time periods, performing one or more physical qubit operations on physical qubits within one or more further synchronisation groups for the third time period defined by the synchronisation scheme, wherein the one or more further synchronisation groups partially overlap the one or more initial synchronisation groups and / or the one or more later synchronisation groups.

7. The method according to any preceding claim, wherein each synchronisation group includes three or more physical qubit subsets.

8. The method according to any preceding claim, wherein each physical qubit subset comprises a plurality of data qubits and a plurality of syndrome qubits.

9. The method according to any preceding claim, wherein each physical qubit subset includes at least a same number of physical qubits as the first and second physical qubit subsets.

10. The method according to any preceding claim, wherein the synchronisation groups partially overlap with one another such that a number of physical qubits common to the partially overlapping synchronisation groups is at least a same number of physical qubits as the first and second physical qubit subsets.

11. The method according to any preceding claim, wherein each synchronisation group comprises physical qubit subsets for which the quantum computing system is capable of performing multi-qubit operations across the physical qubit subsets within the respective time period for the synchronisation group.

12. The method according to any preceding claim, further comprising: prior to performing physical qubit operations in a particular time period, synchronising the physical qubit subsets of each synchronisation group for the respective time period.

13. The method according to claim 12, wherein synchronising the physical qubit subsets comprises synchronising control hardware for performing physical qubit operations on the physical qubits of the respective physical qubit subsets.

14. The method according to any preceding claim, wherein the logical qubit operation comprises a lattice surgery operation.

15. The method according to any preceding claim, wherein the logical qubit operation comprises an entangling operation between the plurality of logical qubits.

16. The method according to claim 15, wherein the logical qubit operating comprises a relocation of the first logical qubit and / or the second logical qubit.

17. The method according to claim 16, wherein the logical qubit operation comprises a relocation of the first logical qubit relative to the second logical qubit.

18. The method according to any preceding claim, wherein the indication of the synchronisation scheme is the indication of the logical qubit operation; and wherein the method further comprises deriving the synchronisation scheme based on the indication of the logical qubit operation.

19. The method according to any preceding claim, wherein the synchronisation groups partially overlap with one another such that some but not all of the physical qubit subsets of a synchronisation group overlap with some but not all of the physical qubit subsets of another synchronisation group of another time period.

20. The method according to any preceding claim, wherein each synchronisation group overlaps with one or more other synchronisation groups for one or more other time periods.

21. The method according to any preceding claim, wherein each synchronisation group includes all physical qubit subsets which are to be synchronised with the respective two or more physical qubit subsets for the respective time period.

22. The method according to any preceding claim, wherein the one or more additional physical qubit subsets comprise physical qubits not included in the physical qubit subsets implementing the plurality of logical qubits.

23. A quantum computing system comprising: a plurality of physical qubits configured to implement a plurality of logical qubits, wherein the plurality of physical qubits includes a plurality of physical qubit subsets, wherein a first logical qubit is implemented by a first physical qubit subset, a second logical qubit is implemented by a second physical qubit subset, and wherein the plurality of physical qubits further includes one or more additional physical qubit; and a control system configured to perform physical qubit operations on the plurality of physical qubits to execute logical qubit operations; wherein the quantum computing system is configured to perform the method of any preceding claim.

24. A computer-readable medium comprising instructions which, when executed by the quantum computing system of claim 23, cause the quantum computing system to perform the method of any of claims 1-22.

25. A computer-implemented method comprising: identifying a logical qubit operation to be performed on a plurality of logical qubits and a quantum computing system on which the logical qubit operation is to be performed; based on the identified logical qubit operation and quantum computing system, identifying a synchronisation scheme for the logical qubit operation, wherein the synchronisation scheme defines a plurality of time periods of the logical qubit operation and synchronisation groups for each of the plurality of time periods, wherein each of the synchronisation groups includes two or more physical qubit subsets of the quantum computing system to be synchronised with one another for a respective time period, and wherein the synchronisation groups partially overlap with one another to form a logical chain of synchronisation groups between physical qubit subsets implementing the plurality of logical qubits, wherein the logical chain of synchronisation groups spans the plurality of time periods; and transmitting an indication of the logical qubit operation and the synchronisation scheme to the quantum computing system.

26. The method according to claim 25, wherein the synchronisation scheme is identified based on a code distance of the plurality of logical qubits.

27. The method according to claim 25 or claim 26, further comprising: transmitting an indication of the logical qubit operation and the synchronisation scheme to a decoder for the quantum computing system.

28. A computer-implemented method comprising: receiving an indication of a logical qubit operation to be performed on a quantum computing system and a synchronisation scheme, wherein the synchronisation scheme defines a plurality of time periods of the logical qubit operation for a plurality of logical qubits implemented by the quantum computing system and synchronisation groups for each of the plurality of time periods, wherein each of the synchronisation groups includes two or more physical qubit subsets of the quantum computing system to be synchronised with one another for a respective time period, and wherein the synchronisation groups partially overlap with one another to form a logical chain of synchronisation groups between physical qubit subsets implementing the plurality of logical qubits, wherein the logical chain of synchronisation groups spans the plurality of time periods; and receiving syndrome data from the quantum computing system for the logical qubit operation, wherein receiving syndrome data comprises: in an initial time period, receiving initial syndrome data for physical qubits within one or more initial synchronisation groups for the initial time period defined by the synchronisation scheme; and in a later time period subsequent to the first time period, receiving subsequent syndrome data for physical qubits within one or more subsequent synchronisation groups for the later time period defined by the synchronisation scheme.

29. The method according to claim 28, further comprising: upon receiving the initial syndrome data, decoding the initial syndrome data; and upon receiving the subsequent syndrome data, decoding the subsequent syndrome data.

30. The method according to claim 28, further comprising: based on receiving both the initial syndrome data and subsequent syndrome data, decoding the initial syndrome data and subsequent syndrome data.

31. A system comprising: a memory; and one or more processors, wherein the one or more processors are configured to perform the method of any of claims 25-30.

32. A computer-readable medium comprising instructions which, when executed by a computer, cause the computer to perform the method of any of claims 25-30.

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