Quantum algorithms for logical qubits in quantum systems with physical qubits that allow for error correction and improved connectivity between qubits

Abstract

This application relates to methods and apparatus for executing quantum algorithms of logical qubits, the quantum algorithms being executed on a quantum system of physical qubits. The method includes providing a quantum system with a plurality of physical qubits in a spatial arrangement. The number of physical qubits is greater than the number of logical qubits. The logical qubits are encoded into physical qubits. The physical qubits represent either the parity of two or more logical qubits or a single logical qubit. A set of physical qubits satisfying a constraint satisfies the constraint, and each constraint-satisfying set includes some physical qubits such that all logical qubits are represented by either 0 or an even number of physical qubits in the set. The method further includes enabling quantum interactions between the logical qubits by performing quantum interactions on the physical qubits. Further, at least some of the physical qubits are measured to obtain a readout.

Description

[Technical Field]

[0001] Embodiments described herein relate to apparatus and methods for executing quantum algorithms, and more particularly to increasing connectivity when executing quantum algorithms of logical quantum bits (qubits) on quantum systems, and more particularly on quantum systems including multiple physical quantum bits, and more particularly, the logical quantum bits are encoded onto the physical quantum bits, and more particularly, the encoding can be used to correct errors that occur on the physical quantum bits. [Background technology]

[0002] Designing quantum computers and quantum algorithms is a current major challenge in science and engineering, motivated by the promise of solving certain problems exponentially faster than known classical algorithms. However, the fundamental rules of quantum mechanics that enable this new methodology also impose fundamental limitations. In contrast to classical information, quantum information cannot be copied—a phenomenon known as the quantum no-cloning theorem—and can only be propagated. Therefore, quantum computers cannot follow a von Neumann architecture with separate memory and computation units. Because quantum CPUs simultaneously function as memory and computation units, connectivity between any qubits on the chip is required. In the current standard approach to gate-based quantum computers, these long-range interactions are implemented as physical interactions, which limit scalability, or quantum information is moved around the chip via a SWAP sequence, which requires significant gate overhead.

[0003] Recent achievements in quantum hardware development on various qubit platforms may soon enable the experimental realization of well-known quantum algorithms for reasonable system sizes. Nevertheless, a fundamental challenge for state-of-the-art quantum devices remains the connectivity between qubits on a quantum chip. This is particularly important because long-range and high-density (ideally all-to-all) connectivity is crucial for many important quantum algorithms unless algorithm-specific preprocessing steps are performed. While several quantum routing techniques exist, this is particularly problematic for device scalability beyond the noisy intermediate-scale quantum (NISQ) era.

[0004] Furthermore, these quantum computing systems are highly complex, and it is difficult to fully implement their hardware systems. In many cases, quantum computing systems introduce errors when executing quantum algorithms that are detrimental to obtaining accurate results of the quantum algorithms. Therefore, there is a strong need to reduce these errors when executing quantum algorithms on quantum systems. Summary of the Invention

[0005] According to one embodiment of the present invention, a method is provided for executing a quantum algorithm on logical qubits on a quantum system. The method includes providing a quantum system with a plurality of physical qubits in a spatial arrangement, the number of physical qubits being greater than the number of logical qubits. The method further includes encoding the logical qubits into physical qubits, where each encoded physical qubit represents either the parity of two or more logical qubits or a single logical qubit, and the physical qubits in a set satisfying the constraint satisfy the constraint, each constraint-satisfying set including several physical qubits such that all logical qubits are represented by zero or an even number of physical qubits in the set, and the physical qubits in a set satisfying the constraint are preferably physically adjacent in the spatial arrangement. The method includes performing a quantum interaction between the logical qubits by performing a quantum operation on the physical qubits. The method further includes measuring at least some of the physical qubits to obtain a readout.

[0006] The present invention's encoding of logical qubits onto physical qubits by encoding their parity allows for increased connectivity of the logical qubits. Therefore, any quantum algorithm can be implemented via operations on the physical qubits, without the need for swap gates and with the possibility of parallelization of quantum gates. Furthermore, the encoding is highly flexible, allowing algorithm-dependent encoding of the physical qubits, reducing resources and minimizing errors. Furthermore, the present invention's encoding provides redundant encoding of the information of the logical qubits. On the one hand, this redundancy makes it possible to perform quantum interactions between any logical qubits by applying corresponding operations to the physical qubits. On the other hand, the redundancy provides an inherent possibility for detecting and correcting errors in the physical qubits.

[0007] According to another embodiment of the present invention, an apparatus for executing a quantum algorithm on logical qubits is provided. The apparatus comprises a quantum system comprising a plurality of physical qubits in a spatial arrangement. It further comprises a quantum operation unit configured to perform a quantum operation on the physical qubits corresponding to a quantum interaction of the logical qubits. The apparatus comprises a measurement device adapted to measure at least a portion of the plurality of physical qubits. The apparatus further comprises a classical computing system connected to the measurement device and / or the quantum operation unit. The apparatus is configured to encode the logical qubits into physical qubits such that each physical qubit represents either the parity of two or more logical qubits or a single logical qubit, and sets satisfying the encoding constraints of the physical qubits satisfy the constraints, each constraint-satisfying set including several physical qubits such that all logical qubits are represented by either 0 or an even number of physical qubits in the set. The physical qubits in a set satisfying the constraints are preferably physically adjacent to each other in the spatial arrangement.

[0008] Further advantages, features, aspects and details that can be combined with the embodiments described herein become apparent from the dependent claims, the description and the drawings. [Brief explanation of the drawings]

[0009] A full and enabling disclosure to one of ordinary skill in the art is set forth more particularly in the remainder of the specification, including reference to the accompanying drawings. [Figure 1] FIG. 1 shows an exemplary encoding having physical qubits arranged in a two-dimensional lattice including data qubits and parity qubits. [Figure 2] FIG. 2 illustrates a procedure for encoding and decoding a physical qubit using a controlled-NOT gate that can be used in embodiments described herein. [Figure 3]FIG. 3 illustrates a procedure for encoding and decoding physical qubits using ancilla qubits, measurement and correction in response to the measurement results, that can be used in the embodiments described herein. [Figure 4] FIG. 4 shows an implementation of error correction for physical qubits using ancilla qubits and measurements on these ancilla qubits that can be used in embodiments described herein. [Figure 5] FIG. 5 shows an implementation of error correction for physical qubits using ancilla qubits and measurements on these ancilla qubits that can be used in the embodiments described herein. [Figure 6] FIG. 6 shows a comparison of implementations of a universal gate set in a standard implementation and an implementation with logical qubits encoded into physical qubits according to the invention. [Figure 7] Figure 7 shows the realization of a logical single-qubit unitary on a physical qubit. [Figure 8] Figure 8 shows a possible implementation layout for quantum addition using physical qubits. [Figure 9] Figure 9 shows the implementation layout of an n-controlled phase gate that uses an additional ancillary qubit. [Figure 10] FIG. 10 shows the gate decomposition of a multi-controlled phase gate using an ancilla qubit. [Figure 11] FIG. 11 shows the gates of a circuit for performing a quantum Fourier transform on physical qubits, where only gates that contribute to the full circuit depth, i.e., cannot be further parallelized, are shown; all other gates can be performed in parallel with the gates shown. DETAILED DESCRIPTION OF THE INVENTION

[0010] Reference will now be made in detail to various exemplary embodiments, one or more examples of which are illustrated in the figures. Each example is provided by way of explanation and not by way of limitation. For example, features illustrated or described as part of one embodiment can be used on or in combination with other embodiments to yield yet a further embodiment. This disclosure is intended to include such modifications and variations.

[0011] In the following description of the drawings, like reference numerals refer to like components. In most cases, only the differences with respect to individual embodiments will be described. The structures shown in the drawings are not necessarily drawn to scale and may include details that are depicted in an exaggerated manner to allow a better understanding of the embodiments.

[0012] According to one embodiment of the present invention, a method is provided for executing a quantum algorithm of logical qubits on a quantum system of physical qubits. A physical qubit is a component of a quantum system. The physical qubits may be arranged in a spatial arrangement such that quantum interactions are possible between physical qubits that are physically close in the spatial arrangement. For example, the physical qubits may be arranged according to a square lattice. In this square lattice, the physical qubits may occupy vertices. In one physical implementation, interactions are possible between qubits on adjacent vertices. In other physical implementations, interactions between next-nearest neighbors, i.e., along vertices that are diagonal to the square lattice, may also be possible. In a more general situation, the qubits may be arranged according to a mesh. This mesh may be a two-dimensional mesh. The vertices of the mesh represent possible locations of physical qubits. In the mesh, cells may indicate that quantum interactions are possible between physical qubits located in that cell. The cells may be two-dimensional.

[0013] The number of physical qubits is greater than the number of logical qubits; for example, n logical qubits can be encoded into K>n physical qubits. That is, the logical qubits are encoded into a larger number of physical qubits. Each physical qubit represents either the parity of two or more logical qubits or a single logical qubit. For example, if the physical qubits are initialized in the z basis, i.e., the basis states of a single qubit are |0〉 and |1〉, then the physical qubit can represent the parity of two logical qubits.

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[0014] Physical qubits are of course not limited to any basis, and the parity of two or more logical qubits, or a single logical qubit, can be represented in any other basis in a manner similar to the above example of a physical qubit initialized in the z basis.

[0015] A set that satisfies the encoding constraints of a physical qubit satisfies the constraint. Each constraint-satisfying set contains an even number of physical qubits that represent the same logical qubit either by 0 or by the parity they have with other logical qubits, or by a direct correspondence thereto. In other words, each constraint-satisfying set contains some number of physical qubits such that all logical qubits are represented by 0 or an even number of physical qubits in the set. Without being bound to any particular theory, the following is offered as an explanation of the importance of sets that satisfy the constraints to quantum algorithms performed on quantum systems.

[0016] Considering the above example, these constraints are:

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[0017] In a preferred embodiment, the physical qubits in a set that satisfies the constraints are physically close together in a spatial arrangement. In the above example of physical qubits arranged on a square lattice, the sets that satisfy these constraints may be formed by physical qubits arranged on adjacent vertices. The sets that satisfy the constraints may form cells, but preferably may also be formed by lines of adjacent qubits.

[0018] Physical quantum operations or interactions are applied to physical qubits to enable quantum interactions between the logical qubits and thus perform quantum algorithms on the states of the logical qubits. Preferably, the physical interactions are applied to physical qubits that are in physical proximity to one another. As described above, the encoding of the present invention allows for encoding logical qubits such that interactions on the logical qubits correspond to operations on physically nearby physical qubits.

[0019] Many quantum algorithms require long-range interactions between qubits. The present invention's encoding of the parity of logical qubits onto physical qubits allows these substantially long-range interactions of logical qubits to be implemented as short-range interactions of physical qubits. Furthermore, the spatial arrangement of physical qubits can be changed in the encoding, allowing for the creation of encodings of physical qubits tailored to specific algorithms. To find these tailored encodings, for example, quantum operation control units such as those described in International Publication No. 2022 / 008057 can be used. The document International Publication No. 2022 / 008057 is incorporated by reference.

[0020] Universal Gate Set In certain embodiments, the quantum operation is an operation from a universal gate set. This universal gate set can include gates corresponding to rotation operator gates and controlled phase gates. This is an example of a universal gate set, but embodiments of the invention are not limited to this example. A rotation operator gate is a single-field unitary operator, an operator that acts on a single qubit, e.g., the i-th qubit, of a quantum system. Mathematically, a single-field unitary operator can be of the form

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[0021] Any single qubit operator on a logical qubit can be constructed by a logical rotation operator gate, for example, any single qubit operation on a logical qubit (i) is

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[0022] As shown in Figure 6, these single-qubit rotation operator gates can be implemented on physical qubits in the following way: A rotation of a logical qubit (i) about the z-axis corresponds to a rotation of the corresponding physical qubit about the z-axis, which means that the logical qubit, e.g.

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[0023] The right column of Figure 6 shows physical qubits arranged in a two-dimensional square lattice, which is an exemplary spatial arrangement for encoding logical qubits into physical qubits. The physical qubits are located at the vertices of the two-dimensional square lattice, forming triangular portions of the two-dimensional square lattice. The squares and triangles connecting the physical qubits correspond to sets that satisfy the constraints.

[0024] As also shown in Figure 6, the controlled phase gates required to obtain a universal gate set that requires a rotation operator gate plus an additional two-qubit entangling gate can be easily implemented on physical qubits via local operations. For example, to implement a controlled phase gate on logical qubits (i) and (j), one need only perform a local rotation about the z axis of three physical qubits: the two physical qubits representing logical qubits (i) and (j), and physical qubit ij representing the parity of the two logical qubits (i) and (j).

[0025] Thus, through this example set of universal gates, any quantum circuit can be constructed, and therefore any quantum algorithm can be executed via the physical qubits. To obtain a result, at least some of the physical qubits are finally measured to obtain a readout. Depending on the algorithm being executed, different qubits can be measured for readout. Because the logical qubit states are redundantly encoded into the physical qubits, in general, various possible groups of qubits can be measured to provide a readout.

[0026] encoding In certain embodiments of the invention, as shown in FIG. 1, physical qubit 100 includes data qubits 111 and / or parity qubits 222, where the number of data qubits 111 is equal to or less than the number of logical qubits, and where data qubit 111 represents a single logical qubit and parity qubit 222 represents the parity of at least two logical qubits. In FIG. 1, physical qubit 100 includes five data qubits 111 labeled by their corresponding logical indices 0 through 5. Furthermore, physical qubit 100 includes ten parity qubits 222 labeled by index ij, representing the parity of two logical qubits, where i ≠ j ∈ {0,...,4}. For example, parity qubit 222 labeled 04 represents the parity of logical qubits 0 and 4. When the state of a logical qubit includes five qubits, the encoding of FIG. 1 is an example of an encoding with physical qubits 100 that include data qubits 111 that correspond directly to the logical qubits and all parity qubits 222 that represent the parity of the two logical qubits (i.e., there is a parity qubit 222 for every combination of the two logical qubits). Thus, in this example, n logical qubits are encoded into n(n+1) / 2 physical qubits 100, e.g., five logical qubits are encoded into 15 physical qubits 100. In other words, as shown in FIG. 1, n all-to-all connected logical qubits can be represented by n(n+1) / 2 physical qubits 100. Furthermore, for all-to-all connectivity, the physical qubits 100 satisfy the n(n+1) / 2-n constraint; e.g., in the encoding of FIG. 1, this corresponds to a constraint of 10. In FIG. 1, sets 101 that satisfy these constraints are highlighted by triangles and squares between corresponding physical qubits 100. A triangle corresponds to a 3-regime constraint between three physical qubits 100 at the corners of the triangle, and a square similarly corresponds to a 4-regime constraint between four physical qubits 100. Note, however, that a set 101 that satisfies the constraint need not correspond to a cell.In different embodiments, physical qubits 100 may be arranged in other spatial arrangements, such that qubits in set 101 that satisfy the constraint may be arranged, for example, on a line.

[0027] 1 depicts three lines 102 connecting five physical qubits 100, one double, one bold, and one dashed. These lines, also referred to as logical lines 102, connect physical qubits 100 that share the same logical index i. In FIG. 1, the double lines correspond to physical qubits 100 that share index 1, i.e., represent logical qubit 1 (data qubits 111, 1) or represent logical qubit 1 and the parity of another logical qubit (parity qubit 222). As described above, a rotation operator, which rotates a single qubit around the x-axis, can be implemented on physical qubits 100 by a series of CNOT gates along logical lines 102.

[0028] In a preferred embodiment, the encoding process encodes the parity of a set of n physical qubits into a new physical qubit. This process is uniquely defined by its action on the computational basis states. For example, if the n physical qubits are initially in states |l1〉|l2〉...|ln〉, where l i ∈ {0,1}, then the encoding adds a new physical qubit (previously separable from the physical qubits in the encoding) and places the combined system in state

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[0029] For example, a new physical qubit encoded from parity qubits 12 and 23 represents parity 13.

[0030] Every encoding process can also have a constraint associated with it. This constraint requires that the new physical qubit preserve the parity of the other physical qubits used in the encoding. In the example above, this means that parity qubits 12, 23, and 13 satisfy the constraint, forming constraint-satisfying set 101. As illustrated in FIG. 1, this constraint-satisfying set 101 may correspond to a triangle connecting parity qubits 12, 23, and 13. Thus, the encoding process guarantees that the constraint is satisfied by the physical qubits in constraint-satisfying set 101.

[0031] This encoding scheme also allows for a very simple decoding scheme for the physical qubits. Thus, in another embodiment of the present invention, the physical qubits are decoded. For the implementation of quantum algorithms, it may be useful or necessary to decode the physical qubits, for example, the physical qubits representing logical ancillary qubits. Decoding is performed by a reverse encoding process. In the decoded state, the removed physical qubit is always free from entanglement with all other physical qubits. An important advantage of this encoding and decoding is that the amount of parity qubits can be changed by decoding or encoding, which adds or removes new parity qubits from the physical qubits.

[0032] Preferably, in a first encoding step, the data qubits are initialized with the quantum state of the logical qubit, and in a further step, additional physical qubits are initialized and new parity qubits are added to the encoding by encoding the parity of some of the physical qubits into new parity qubits to form a set that satisfies the constraint with the new parity qubits. Furthermore, it is possible to initialize further data qubits in the encoding scheme after the parity qubits have been added to the encoding.

[0033] A possible way to perform encoding is to initialize a new physical qubit in state |0〉 and apply physical CNOT gate 441 to each of the existing physical qubits for which the new physical qubit will represent parity, i.e., each of the physical qubits that form new parity qubit 222 and constraint-satisfying set 101 (see Figure 2). CNOT gate 441 forces the new physical qubit to have the correct parity because the new physical qubit is bit-flipped with respect to all control qubits that correspond to other physical qubits in the same cell as the new physical qubit, which are in state |1〉. Decoding is the reverse process and can therefore be performed with the reverse gate sequence. In the example of Figure 2, physical qubit 03, which is parity qubit 222, is encoded or decoded by applying CNOT gate 441 to this new physical qubit and to physical qubits 02, 12, and 13, which represent the parity of logical qubits 0 and 3. Thus, physical qubits 02, 12, 13, and 03 form set 101 that satisfies the constraints shown as squares in Figure 2. After decoding, the previous physical qubit 03 is separated from all physical qubits and is shown as an open circle in Figure 2.

[0034] For decoding larger sets of physical qubits, the order in which they are decoded can be important. Physical qubits removed from the encoding cannot be used to later decode other physical qubits. For example, if parity qubit 01 is decoded and then parity qubit 02 must also be decoded, parity qubits 01 and 12 can no longer be used because 01 has already been decoded. Therefore, other physical qubits, which may be further away in the spatial arrangement, such as data qubits 0 and 2, according to the mesh, must be used. However, if parity qubit 02 is decoded first, adjacent parity qubits 01 and 12 can be used for decoding. Subsequently, adjacent data qubits 0 and 1 can be used to decode parity qubit 01.

[0035] In a different embodiment of the encoding, each new parity qubit 222 is initialized in the state |+〉, a constraint measurement 445 is applied to the ancilla qubit 333 initialized in the state |0〉, and a controlled NOT gate 441 is applied to the ancilla qubit from each of the physical qubits that form the set 101 that satisfies the new parity qubit 222 and the constraint. Depending on the result of the constraint measurement 445, a bit-flip operation 444 is either applied or not to the new parity qubit 222. This decoding and encoding scheme is shown in FIG. 3 as a simple example. This embodiment is an alternative method for adding physical qubits to the encoding via a measurement-based approach, which is useful when many physical qubits are being encoded or decoded at once. In this embodiment, measurements and classical corrections depending on the measurement results are used. Each new physical qubit 100 to be added to the encoding is initialized in the state |+〉. Next, one can select constraints that fix the new qubit relative to physical qubits, e.g., parity qubits or data qubits 222, 111, that, after encoding, form a set 101 that satisfies the constraint with the new parity qubit 222. If the physical qubits are arranged according to a mesh, the physical qubits may be arranged in the same cell as the new physical qubit. The constraint value is measured using an ancilla qubit 333 that is initialized with the state |0〉 and that correlates with the physical qubits in the set 101 that satisfies the constraint and the new physical qubit. This correlation can be achieved by applying a CNOT gate 441 from each of the physical qubits in the constraint to the ancilla qubit 333, with the physical qubit acting as the control qubit for the CNOT gate 441. It is always possible to define such new constraints for all new parity qubits 222, thereby obtaining sets 101 that satisfy the new constraints associated with them. For example, it can be done using the operator |0〉, which connects the new parity qubit 222 to adjacent physical qubits in, for example, the cells of a two-dimensional rectangular mesh.

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[0036] In the example of Figure 3, physical qubit 03 is assumed to be re-encoded by applying a CNOT gate 441 to ancilla qubit 333 and physical qubits 02, 12, and 13. The combined ancilla qubit 333 is then measured in the z basis 445, and depending on the result, the constraint

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[0037] For decoding, the constraints are no longer needed since they have been satisfied for the physical qubits. Thus, the physical qubit to be decoded is measured directly in the X basis 442, and a phase reversal (Z) operation 443 is applied to the set of physical qubits whose parity is described by the physical qubit to be decoded, i.e., the set of physical qubits in set 101 that satisfy the constraints of the physical qubit to be decoded. This can be understood as a classical analogue of the CNOT gate sequence in the above embodiment.

[0038] This measurement-based strategy can also be implemented at a constant depth to encode and decode multiple physical qubits that may depend on each other. The advantage of the measurement-based strategy is that all gates and measurements can be applied in parallel. The correction required after a measurement may then depend on the results of multiple measurements. Every constraint measurement involving two or more new physical qubits also depends on whether some of these new physical qubits should have been inverted by other constraint measurements. As in the constant-depth embodiment, this inversion occurs only after all constraint measurements, so the results of all measurements must be reinterpreted taking into account all correction inversions. To this end, a set S can be defined as the set of physical qubits for which the required corrections have been determined. Initially, S includes all physical qubits that do not receive any corrections. In the encoding process, this is precisely the physical qubits that are already part of the encoding. In the decoding process, this includes some of the physical qubits to be decoded (those on which other physical qubits do not depend via decoding constraints), and may also include physical qubits that remain encoded (physical qubits that are not in any of the decoding constraints). The following steps can be applied until all physical qubits are present in S. 1. Given the determined corrections, identify corrections for all physical qubits in S such that the corrections depend only on the results of measurements of the physical qubits in S. 2. Update S (add the physical qubit determined in step 1 to S).

[0039] Error Correction Methods according to embodiments described herein include performing error correction on physical qubits. Additionally, apparatus according to embodiments described herein include a quantum operation unit, a measurement device, and a classical computing system adapted to perform error correction. Error correction may be performed by detecting errors via constraint measurements, i.e., measurements of the parity of physical qubits in a set that satisfy a constraint, and applying a local operation to the physical qubit in which the error was detected according to the results of the constraint measurements.

[0040] In the apparatus and method according to the present invention, redundant encoding of logical qubits onto physical qubits makes it possible to detect and correct errors occurring on the physical qubits. If each constraint is satisfied by the physical qubits in the set that satisfy the constraint, measuring their values ​​does not disturb the logical state of the system, since the constraints commute with any logical operator. Therefore, constraint measurements can be used for error correction. The constraint measurements required to detect errors can be implemented, for example, with the help of an ancilla qubit. For constraint measurements, the ancilla qubit is first coupled to all physical qubits in the set that satisfy the constraint, for example, by applying a CNOT gate from the physical qubits that satisfy all constraints to the ancilla qubit. After coupling, the ancilla qubit is measured, and a readout is obtained. With the help of the measurement, it is possible to determine whether the constraint is satisfied by the physical qubits in the set that satisfy the constraint to which the ancilla qubit is coupled. This is one exemplary embodiment of constraint measurements, but these measurements can be achieved in many other ways.

[0041] In exemplary embodiments, the physical qubits may be arranged according to a mesh, i.e., the physical qubits may be arranged at the vertices of the mesh, and the sets that satisfy the constraints may correspond to the cells of the mesh. In such embodiments, the ancilla qubit may be arranged, for example, at the center of a cell containing physical qubits whose vertices correspond to the sets that satisfy the constraints. The physical qubits of the sets that satisfy the constraints may be arranged according to all or some of the vertices of the cells of the mesh. The ancilla qubits may then be coupled to all physical qubits in the cells that satisfy the constraints. In particular embodiments, this coupling may be performed by applying a CNOT gate from each of the physical qubits in the sets that satisfy the constraints to the ancilla qubit, with the physical qubit acting as the control qubit. The ancilla qubit may then be measured in the Z basis. The measurement result indicates whether the constraints are still satisfied by the physical qubits or whether an error, i.e., a bit-flip error, has occurred.

[0042] To find which physical qubit error occurred, several constraint measurements are applied. Preferably, the same constraint measurement for a group of physical qubits is applied multiple times to tolerate measurement errors. After all constraint measurements have been performed, a decoding algorithm can be used to determine the most likely error scenario from the measured data. The decoding algorithm can be implemented with the help of a classical computing system. The measurement data includes information for all constraint measurements, whether the constraint is satisfied or not. The most likely error scenario indicates which physical qubit and at what point in time the error occurred. There are many different decoding algorithms that can be used to determine the error scenario, one example being belief propagation.

[0043] FIG. 4 shows an illustrative example, illustrating how it is possible to detect which physical qubit 100 has experienced a single error 500 from the results of a constraint measurement 445. In this example, the constraint is measured only once, and the measurement result is assumed to be correct and / or reliable. Furthermore, the physical qubits 100 are arranged at the vertices of a two-dimensional square lattice, and a set 101 of physical qubits 100 that satisfies the constraint is marked with a square connecting four physical qubits 100. In this example, an error 500, e.g., a bit-flip error, occurred on a physical qubit 100 marked with a white cross. Due to this error 500, the four constraints 101 are no longer satisfied, which can be detected, for example, by a constraint measurement 445 on an ancilla qubit 333 coupled to the corresponding physical qubit 100. An example of a constrained measurement 445 via an ancilla 333 is shown in FIG. 5, where the ancilla qubit 333 is placed in the center of the constraint-satisfying set 101 and coupled to all physical qubits 100 in the constraint-satisfying set 101 by applying a CNOT gate 441 from each of the physical qubits 100 in the constraint-satisfying set 101 to the ancilla qubit 333. After this coupling, the ancilla qubit 333 can be measured in the z basis 445 to determine whether the constraint is satisfied by the physical qubits 100. These constrained measurements 445 indicate that for the example of FIG. 4, the most likely error scenario is one in which an error 500 occurs in the marked physical qubit. All other error scenarios that result in the same constrained measurement outcome are much less likely because many physical qubits 100 would be flipped simultaneously in these scenarios. This is highly unlikely if the average error rate on the physical qubits 100 is sufficiently small. Information about the most likely error scenarios can be obtained with the help of classical decoding algorithms. A flipped physical qubit can be corrected by applying a local bit-flip operation.

[0044] In another embodiment of the present invention, the physical qubit is a noise-bias qubit. With the aid of a constraint measurement, one type of error, e.g., a bit-flip error, can be detected. Therefore, the inventive encoding of the physical qubit allows for the correction of one type of error. To complement this ideally, the physical qubit can be a noise-bias qubit that is inherently robust against other types of errors, e.g., phase-flip errors. Such noise-bias qubits are, for example, resonator cat qubits or asymmetric GKP qubits. These noise-bias qubits have the ability to suppress one type of noise at the expense of increasing the proportion of the other type of noise. The inventive redundant encoding of the physical qubit allows for the correction of the other type of noise, ultimately reducing all types of errors.

[0045] Devices and Quantum Systems An embodiment of the present invention comprises an apparatus for executing quantum algorithms on logical qubits, i.e., on which the method according to the present invention can be implemented.

[0046] The apparatus and method according to the present invention comprise a quantum system having a plurality of physical qubits in a spatial arrangement. The quantum system may be, for example, a superconducting circuit, a system of one or more ion traps, a system of neutral atoms, a system of photons, a spin system, etc. The apparatus may include general-purpose quantum computers and classical computing systems, allowing any quantum algorithm to be performed using the apparatus. In certain embodiments, the components of the quantum system, i.e., qubits, may be in a particular spatial arrangement in which quantum interactions are only possible between nearest neighbors and / or next-nearest neighbors. Due to the present encoding of logical qubits, this is not a limitation for the present methods and apparatus, since interactions between logical qubits can be implemented by neighboring physical qubits. However, it is important to emphasize that the methods and apparatus are independent of the specific quantum system. Therefore, any available quantum system containing qubits can be used. Quantum systems containing qubits, i.e., d-level systems, can also be used, since it is always possible to address only two levels of these d-level systems. Furthermore, as described above, the physical qubits may be noise bias qubits to minimize errors occurring during execution of the quantum algorithm. In another embodiment, the physical qubits are arranged on a chip. For example, the physical qubits may be arranged according to a two-dimensional square lattice on the chip.

[0047] The apparatus includes a quantum operation unit configured to perform quantum operations on physical qubits corresponding to quantum interactions on logical qubits. The apparatus further includes a measurement device adapted to measure at least a portion of the plurality of physical qubits, and a classical computing system connected to the measurement device and / or the quantum operation unit. With the aid of these units and systems, the apparatus is configured to encode logical qubits into physical qubits such that each physical qubit represents either the parity of two or more logical qubits or a single logical qubit, where sets satisfying encoding constraints for the physical qubits satisfy the constraints, each constraint-satisfying set includes several physical qubits such that all logical qubits are represented by either 0 or an even number of physical qubits in the set, and the physical qubits in a set satisfying the constraints are preferably physically close to each other in a spatial arrangement. In this way, the apparatus enables the encoding of the present invention. As described above, there are several possible options for encoding the parity of logical qubits into physical qubits. Therefore, the apparatus can be adapted according to the method used to encode the physical qubits. While certain embodiments may use only a quantum operation unit to encode the logical qubits into physical qubits, in other embodiments, both the measurement device and the classical computing system may be used for encoding. The classical computing system may be connected to both the measurement device and the quantum operation unit, e.g., allowing for performing quantum operations on physical qubits in response to results of measurements performed by the measurement device. The classical computing system may also include a controller configured to control the measurement device and / or the quantum operation unit.

[0048] The quantum operation unit enables the implementation of quantum algorithms by implementing interactions between physical qubits. The physical qubits in a set that satisfies the constraints may be physically close to each other, for example, arranged within a cell of a mesh in which the physical qubits of a quantum system are arranged. In the encoding, the physical qubits may be spatially arranged so that the physical qubits to which the quantum interaction applies are physically close to each other. Therefore, the arrangement of the physical qubits in the encoding can be adapted to the algorithm implemented via the quantum system. Therefore, the encoding of the present invention avoids the need for long-range interactions, which require the implementation of, for example, a SWAP gate, which is highly error-prone and requires significant overhead for gates, particularly quantum interactions. Therefore, the method and apparatus of the present invention can be implemented on a variety of current noisy intermediate-scale quantum (NISQ) devices due to the increased connectivity between logical qubits.

[0049] Furthermore, the measurement device of the present apparatus is a standard measurement device that allows measuring qubits and obtaining readouts, depending on the type of quantum system used. Measurement of physical qubits, such as ancillary qubits, may also be required during encoding or error correction.

[0050] The apparatus also includes a classical computing system connected to the measurement device and / or the quantum computing unit. The classical computing system, for example, enables processing of the measurement results. A classical computing system can refer to a computing system that operates on bits or other classical units of information. A classical computing system can include a central processing unit (CPU) for processing information represented by the bits and / or a memory for storing information represented by the bits. A classical computing system can include one or more conventional computers, such as personal computers (PCs), and / or a network of conventional computers.

[0051] In another embodiment of the invention, the apparatus is also adapted to detect and correct errors, the measurement device is adapted to perform constraint measurements on the physical qubits in the set that satisfy the constraint to obtain an error syndrome, the classical computing system is adapted to receive and decode the measured error syndrome and determine the actual error to be corrected, and the quantum operation unit is adapted to apply an operation to the subset of the physical qubits according to the decoded error syndrome. The constraint measurements can be performed as described above.

[0052] Furthermore, the apparatus may also include an initialization unit configured to initialize the physical qubits in the quantum states. For example, the initialization unit may allow for initializing data qubits in the quantum states of logical qubits. Furthermore, depending on the encoding method used, this may allow for initializing ancillary qubits for encoding or error correction of a particular quantum state, and parity qubits to be added to the encoding of a particular quantum state. It should be noted that for certain quantum systems, initialization may not be necessary if the qubits of the quantum system are capable of self-initializing themselves.

[0053] The following describes an example of a quantum system including neutral atoms and its implementation for methods and apparatus according to the present invention. The following is merely an example of a quantum system, and the embodiments described herein are not limited to this exemplary implementation.

[0054] To provide neutral atoms (alkali or alkaline earth species), for example, rubidium atoms (87Rb) can be arranged in a two-dimensional optical tweezers array.

[0055] The computational state of the qubit in this system can be in two different hyperfine basis states, e.g., F=1 and F=2 (requiring the presence of a magnetic field).

[0056] The physical qubits are initialized by optical pumping, and therefore the initialization unit is configured to perform optical pumping.

[0057] For readout with the measurement device, a fluorescence image of the tweezers array or region of interest is acquired, whereby atoms in state |0〉 appear bright, while atoms in state |1〉 remain dark.

[0058] The quantum operation unit allows for the implementation of single-qubit and multi-qubit gates, which may be implemented as described below.

[0059] Single qubit gates: Single qubit gates are realized by two simultaneous laser pulses, the frequency of which is adjusted so that the atom being treated interacts with the electromagnetic field of the laser, which is spatially focused at the location of the atom being treated.

[0060] Multi-qubit gates: A controlled Z gate between two atoms can be implemented by driving both atoms with a laser that couples their states through a highly excited Rydberg state (in the Rydberg cutoff, where two neighboring atoms cannot be simultaneously excited to a Rydberg state due to strong dipole-dipole interactions). All other multi-qubit gates, such as the controlled NOT gate, can be derived from this gate by applying additional single-qubit gates. While these gates can, in principle, be implemented between any pair of qubits due to the strong long-range dipole-dipole interactions, the closer the qubits are, the better the gate fidelity and / or the faster the implementation time (because the interaction has a power-law decay behavior in distance). Gates between nearest and next-nearest neighbors on a square lattice can be implemented with sufficient fidelity.

[0061] Applications and examples of connectivity improvements in known quantum algorithms The scope of application of the above-described universal gate set in combination with the inventive method and apparatus for increasing connectivity between qubits in the described encoding of logical qubits is demonstrated below by exploring several quantum gate algorithms, such as quantum Fourier transform and quantum addition. Embedding these algorithms in the encoding reduces the circuit depth compared to traditional gate-based implementations while keeping the number of multi-qubit gates comparable. Note that all mentioned algorithms are merely examples of using the inventive encoding of logical qubits to physical qubits and can also be used with any other algorithms.

[0062] A universal gate set may be provided by an encoding that includes all possible data qubits. A universal gate set is defined by the operator

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[0063] 1. Arbitrary single-qubit gates Any single-qubit unitary U can be rotated U =

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[0064] Furthermore, the single qubit R x The rotation can be performed on a physical qubit, the parity qubit, that represents the parity of two logical qubits instead of the data qubits of the logical line. In that case, the decomposed rotation cannot be recombined, but for n>4, an additional R zThe rotation can be performed in parallel with a chain of CNOTs. Combined with the partial parallelization of the chain of CNOTs, this results in a minimum circuit depth of 0.01 for any single-qubit unitary.

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[0065] 2.2 qubit gates logic

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[0066] 3. Quantum Fourier Transform The quantum Fourier transform (QFT) is the quantum analogue of the discrete Fourier transform and is a key component of many quantum algorithms, such as quantum phase estimation and even Shor's prime factorization algorithm. The unitary function for the QFT for n qubits is

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[0067] 4. Quantum Addition Another prominent problem in various quantum algorithms is quantum addition, a fundamental component of algebraic operations on quantum registers. An efficient circuit for the addition of two quantum registers based on QFT performs the addition in Fourier space using conditional rotation gates.

[0068] Quantum addition (modulo 2n) of two n-qubit quantum states stored in registers R1 and R2 is performed by performing a QFT on one register (e.g., register R1) and then performing a control R between the two registers. z This is achieved by performing a rotation and finally applying the inverse QFT to the previously Fourier transformed register.

[0069] On a chip with everything and everything connected, the quantum addition algorithm can be implemented logarithmically, ignoring the gates required to perform the necessary QFT steps. Our encoding allows the logarithmically deep portion to be performed in a single step, without the need for swap gates, and allows the addition of linear-depth QFT circuits.

[0070] As mentioned above, a control rotation can be implemented with only a single-qubit operation, provided the necessary physical qubits are available. A spatial arrangement of physical qubits that satisfies this requirement is the two-dimensional square lattice shown in Figure 8. The number of physical qubits increases by n(n+1) compared to previously introduced architectures, resulting in a total of 3n(n+1) = 2 qubits. If one of the registers does not require any computation within itself (e.g., if it represents classical data), the corresponding physical qubit for the interaction within that register is not required, and the register's data qubit can be added directly to the continuation of the remaining logical lines, requiring a total of n(n+2) qubits. Note that either way, the expansion of the logical lines has little impact on the circuit depth of the QFT portion, since they do not intersect with any lines corresponding to the other registers. Thus, the encoding allows the central quantum addition circuit to be executed in a single pass, while the surrounding QFT portions have a depth linear in n.

[0071] 5. Multi-control gate and Grover diffusion operators A difficulty that arises in many quantum algorithms is the implementation of a multi-controlled quantum gate because it requires a significant number of nonlocal interactions, and therefore a SWAP gate. Below is an implementation of an m-controlled phase gate in an encoding with m+1 logical qubits. In Grover's search algorithm, multi-controlled phase gates are particularly relevant for implementing the diffusion operator. The diffusion operator for m+1 qubits corresponds to an m-controlled phase gate. An m-controlled phase gate can be decomposed into a 2-controlled phase gate and a Toffoli gate. The decomposition is shown in Figure 10, which introduces m-1 ancillary qubits labeled with capital letters, where m is the number of control qubits. In encoding, this allows for the implementation of an m-controlled phase gate with gate resources that scale linearly with m, introducing a total of 4m+3 ancillary qubits. Furthermore, it is not necessary to use all physical qubits if the corresponding connections are not required. Figure 9 shows a possible layout with physical qubits for implementing a diffusion operator with m=4.

[0072] 5. Optimization Problems The optimization problem is of the form

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[0073] 6. Preparing the graph state Graph states are a key resource for measurement-based quantum computing and some error-correction protocols: cluster states corresponding to square lattice graphs can be efficiently prepared using nearest-neighbor interactions.

[0074] However, preparing a more arbitrary graph state typically involves applying CZ gates to every edge in the graph, requiring long-range interactions between many different qubits. This encoding allows the creation of any graph state containing n qubits with a circuit depth of up to n+3 and 2n(n-1) CNOT gates, as follows: First, n data qubits in superposition state |+〉 are prepared. Next, the encoding sequence is applied in either its full version with all possible physical qubits or its reduced version, depending on the connectivity requirements of the respective graph, requiring a circuit of depth n+1 or less (for the reduced version). Subsequently, CZ gates are applied to all pairs of logical qubits connected in the graph, using parallel single-qubit gates on the physical qubits. The result is the encoded graph state. For measurement-based quantum computing, which involves measuring logical qubits (along any axis), an additional decoding step of depth n+1 is required. The number of physical qubits required for this procedure is, in principle, equal to the number of edges in the desired graph state. However, in some cases it may be useful to have additional ancillary qubits to simplify the encoding and decoding circuits.

[0075] A summary of techniques for implementing fundamental quantum algorithms for arbitrary system sizes is presented, avoiding swap gates entirely. Gate-count analysis shows that this implementation has the potential to save multi-qubit gates or reduce the circuit depth of key components of Shor's algorithm.

[0076] Furthermore, due to the redundant coding of the information, the coding provides an inherent possibility of detecting and correcting errors.

[0077] Furthermore, the encoding only requires interactions between nearest neighbors and can therefore be implemented on a variety of current NISQ devices with their natural inter-qubit connectivity.

[0078] The methods and apparatus of the present invention are also independent of the particular qubit platform. Suitable platforms are, for example, superconducting qubits, neutral atoms, or trapped ions. To complement the error tolerance of the encoding, the use of noise bias qubits may be considered.

Claims

1. 1. A method for executing a quantum algorithm of logical qubits on a quantum system of physical qubits, the method comprising: a. providing a quantum system comprising a plurality of physical qubits in a spatial arrangement, wherein the number of physical qubits is greater than the number of logical qubits; b. encoding the logical qubits into physical qubits, where a physical qubit represents either the parity of two or more logical qubits or a single logical qubit, and the physical qubits in a constraint-satisfying set satisfy the constraint, and each constraint-satisfying set includes a number of physical qubits such that all logical qubits are represented by 0 or an even number of physical qubits in the set; c. enabling quantum interactions between logical qubits by performing quantum operations on physical qubits; d. measuring at least a portion of the physical qubits to obtain a readout; A method comprising:

2. 10. The method of claim 1, wherein the physical qubits in a set that satisfies a constraint are physically located in close proximity in the spatial arrangement.

3. The method of claim 1 or 2, wherein the physical qubits are arranged in a square lattice.

4. 10. The method of any preceding claim, wherein the quantum interactions between physical qubits are performed by applying operations from a set of universal gates.

5. 10. The method of any preceding claim, wherein the method comprises detecting errors via constraint measurements on physical qubits in a set that satisfies a constraint, and performing error correction on the physical qubits by applying an operation on a subset of the physical qubits in response to a result of the constraint measurement.

6. 10. The method of any preceding claim, wherein the physical qubits comprise data qubits and / or parity qubits, the number of data qubits being the same or fewer than the number of logical qubits, the data qubit representing a single logical qubit and the parity qubit representing the parity of at least two logical qubits.

7. 7. The method of claim 6, wherein in the encoding, the data qubits are initialized in the quantum state of the logical qubit, additional physical qubits are initialized, and parity qubits are added to the encoding, and new parity qubits are added by encoding into the new parity qubit the parity of some physical qubits that form a set that satisfies a constraint with the new parity qubit.

8. 8. The method of claim 7, wherein the additional physical qubit is initialized in state |0>, and a new parity qubit is added to the encoding by applying a controlled-NOT gate to the new parity qubit from some of the physical qubits that form a set that satisfies a constraint with the new parity qubit.

9. 8. The method of claim 7, wherein the additional physical qubits are initialized in a state |+〉, the physical qubits including an ancilla qubit initialized in the state |0〉, and a new parity qubit is added to the encoding by applying a controlled-NOT gate to an ancilla qubit initialized from some of the physical qubits that form a set that satisfies a constraint with the new parity qubit, and to the new parity qubit, followed by applying a measurement to the ancilla qubit, and a bit-flip operation is applied or not applied to the new parity qubit depending on the result of the measurement.

10. 10. The method of any preceding claim, wherein the physical qubit is a noise bias qubit.

11. 10. The method of any preceding claim, wherein n logical qubits are encoded into n(n+1) / 2 physical qubits.

12. 10. The method of any one of the preceding claims, wherein the quantum algorithm is an algorithm selected from the following algorithms: Shor algorithm, Grover algorithm, Quantum Approximation Optimization Algorithm (QAOA), Variational Quantum Eigensolver (VQE) algorithm, Quantum Fourier Transform (QFT), quantum addition, Harrow-Hassidim-Lloyd (HHL) algorithm, graph state preparation.

13. 10. The method of any preceding claim, wherein the physical qubits include ancilla qubits, and physical operations and measurements are applied to the ancilla qubits during encoding of the logical qubits onto the physical qubits, and / or during performance of physical interactions between the logical qubits on the physical qubits, and / or during error correction on the physical qubits.

14. 1. An apparatus for executing a quantum algorithm on logical qubits, comprising: a quantum system comprising a plurality of physical qubits in a spatial arrangement; a quantum operation unit configured to perform quantum operations on physical qubits corresponding to quantum interactions on said logical qubits; a measurement device adapted to measure at least a portion of the plurality of physical qubits; a classical computing system connected to the measurement device and / or the quantum processing unit; The apparatus is configured to encode logical qubits onto physical qubits such that each physical qubit represents either a parity of two or more logical qubits or a single logical qubit, wherein in the encoding, a constraint-satisfying set of physical qubits satisfies a constraint, and each constraint-satisfying set includes some physical qubits such that all logical qubits are represented by 0 or an even number of physical qubits in the set.

1. An apparatus comprising:

15. 15. The apparatus of claim 14, wherein the physical qubits in a set that satisfies a constraint are physically located closely together in the spatial arrangement.

16. 16. The apparatus of claim 14 or 15, wherein the apparatus further comprises an initialization unit adapted to initialize some or all of the physical qubits in a quantum state.

17. 17. The apparatus of claim 14, wherein the apparatus is adapted to detect and correct errors, the measurement device is adapted to perform constraint measurements on physical qubits in a set that satisfy a constraint to obtain an error syndrome, the classical computing system is adapted to receive and decode the measured error syndrome and determine the actual error to be corrected, and the quantum operation unit is adapted to apply an operation to a subset of the physical qubits according to the decoded error syndrome.

18. 18. The apparatus of claim 14, wherein the physical qubit is a noise bias qubit.

19. 19. The apparatus of claim 14, wherein the physical qubits are located on a chip.

20. 20. The apparatus of any of claims 14 to 19, wherein the system is one of a superconducting circuit, a system of one or more ion traps, a system of neutral atoms, a system of photons, or a spin system.

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