Decoding quantum error-correcting codes using transformer neural networks

Abstract

A transformer neural network-based decoder for decoding quantum error correcting codes (QECC) includes an input layer, multiple decoding layers, and an output layer. The input layer is adapted to receive an initial noise estimate, calculated by a noise estimator, of noise injected into syndrome bits of a codeword encoded using QECC and transmitted over an interference-affected transmission channel, and to create an embedding of the syndrome bits. Each decoding layer, adapted to calculate an estimated logical operator matrix for each codeword, includes a self-attention layer constructed according to a mask indicating a relationship between embeddings derived from a parity check matrix of the error correcting code. The multiple decoding layers are trained using a joint loss function that aims to minimize the LER, the BER, and the error rate of the noise estimator. The output layer is adapted to generate a vector representing a predicted soft error in the logical operator matrix of the codeword.

Description

[Technical Field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS This application claims the benefit of priority to U.S. Provisional Patent Application No. 63 / 440,625, filed January 23, 2023, the contents of which are incorporated herein by reference in their entirety. [Background technology]

[0002] Some embodiments of the present invention relate to training and using neural networks to decode quantum error-correcting code words transmitted over transmission channels susceptible to interference, and more particularly to training and using decoders based on Transformer neural networks to quantum-decode error-correcting code words transmitted over transmission channels susceptible to interference.

[0003] The transmission of data over transmission channels (either wired and / or wireless) is an essential component of most modern data technology applications, such as, for example, computing platform interconnects (e.g., wafer fabrics, switched fabrics, etc.), memory interfaces, communication channels, network links, etc. Such transmission channels are often, but not always, subject to interference, such as, for example, noise, crosstalk, attenuation, etc., which can degrade the transmission channel and result in data loss at the receiving end.

[0004] To overcome data loss due to such line degradation, error correction information may be added to enable the receiver to detect and correct errors in the received encoded data. Such methods may utilize one or more error correction codes (ECC) and / or models, as known in the art.

[0005] Quantum computing is constantly evolving and attracting attention in industry due to its potential to dramatically improve computational performance. However, quantum computing and data transfer, due to its unique mechanics, may present new challenges not previously encountered with traditional information technology, including the need to support robust data transfer over noisy channels.

[0006] To overcome this challenge, quantum error correction codes (QECCs) can be used, which, like ECCs, add error correction data that can be used to detect and correct errors induced in the transmitted data by noise injected into the transmission channel. Summary of the Invention [Problem to be solved by the invention]

[0007] It is an object of the present invention to provide a method, a system and a software program product for decoding quantum error correcting codes using neural networks, in particular transformer neural networks. These and other objects are achieved by the features of the independent claims. Further embodiments are evident from the description of the dependent claims and the drawings.

[0008] According to a first aspect of the present invention, there is provided a Transformer Neural Network-based decoder for decoding quantum error correcting codes, comprising an input layer, multiple decoding layers, and an output layer. The input layer is adapted to receive initial noise estimates calculated by a noise estimator of noise injected into syndrome bits of one or more codewords encoded using quantum error correcting codes and transmitted over an interference-affected transmission channel, and to create embeddings for the syndrome bits. The multiple decoding layers are adapted to calculate an estimated Boolean operator matrix for each codeword. Each of the multiple decoding layers includes a self-attention layer including one or more heads constructed according to masks indicating relationships between the embeddings, the relationships between the embeddings being derived from a parity check matrix of the error correcting code such that the masks are adapted to unmask connected parity bit pairs and mask disconnected parity bit pairs. The multiple decoding layers are trained using a joint loss function aimed at minimizing the logical error rate (LER), bit error rate (BER), and error rate of the noise estimator. The output layer is adapted to generate a vector representing the predicted soft errors of said logical operator matrix for each codeword.

[0009] According to a second aspect of the present invention, there is provided a method of training a Transformer Neural Network based decoder for decoding quantum error correcting codes, comprising using one or more processors: obtaining a plurality of training samples, the training samples comprising a plurality of initial noise estimates calculated by one or more noise estimators, of noise injected into syndrome bits of one or more codewords encoded with one or more quantum error correcting codes transmitted over one or more transmission channels subject to interference; -A decoder based on a Transformer neural network calculating an estimated Boolean operator matrix for each codeword by minimizing a joint loss function that aims to minimize a logic error rate (LER), a bit error rate (BER), and an error rate of the noise estimator; and by generating a vector representing predicted soft errors in the logical operator matrix of the codeword; using the plurality of training samples to train the Transformer neural network-based decoder to decode codewords encoded with the one or more quantum error correcting codes; - outputting a decoder based on the trained Transformer neural network for decoding one or more codewords encoded with one or more quantum error correcting codes.

[0010] According to a third aspect of the present invention, there is provided a method of using a Transformer Neural Network based decoder for decoding a quantum error correcting code, comprising the steps of: receiving an initial noise estimate, calculated by a noise estimator, of noise injected into syndrome bits of one or more codewords encoded with a quantum error correction code and transmitted over an interference-stricken transmission channel; - applying a decoder based on a trained neural network to calculate an estimated logical operator matrix for said one or more codewords and to generate a vector representing predicted soft errors in the logical operator matrix; - outputting a vector representing the predicted soft errors; The decoder based on the trained neural network is constructed with an input layer adapted to receive the initial noise estimates and create embeddings of the syndrome bits, a plurality of decoding layers adapted to calculate an estimated Boolean operator matrix for the one or more codewords according to a mask indicating relationships between the embeddings, and an output layer adapted to generate a vector representing the predicted soft errors.

[0011] In a further implementation of the first, second, and / or third aspects, the mask is created based on an extended bipartite graph representation of the parity-check matrix of the error-correcting code, the bipartite graph representation including a plurality of nodes connected via a plurality of edges, each pair of connected bits including a bit that shares one or more of the plurality of nodes, and each pair of unconnected bits including a bit that does not share any of the plurality of nodes.

[0012] In a further implementation of the first, second and / or third aspect, the bipartite graph is a Tanner graph.

[0013] In a further implementation of the first, second and / or third aspects, each of the plurality of decoding layers further includes a feedforward layer interleaved with a self-attention layer through a normalization layer.

[0014] In a further implementation of the first, second and / or third aspects, the embedding produced by the input layer has a higher dimensionality than a dimensionality of the received initial noise estimate.

[0015] In a further implementation of the first, second and / or third aspects, the output layer is configured to reduce the dimension of a soft error vector concatenated with multiple soft error vectors calculated by the multiple decoding layers based on the embedding.

[0016] In a further implementation of the first, second and / or third aspects, each of the combined loss functions is adjusted according to a weighting assigned to each of the LER, the BER and the error rate of the noise estimator.

[0017] In a further implementation of the first, second and / or third aspects, the parity check matrix includes a bit-flip parity check matrix calculated to correct bit-flips of quantum bits, and a phase-flip parity check matrix calculated to correct phase-flips of quantum bits.

[0018] In a further implementation of the first, second and / or third aspects, the noise estimator is implemented using one or more shallow neural networks parameterized during training using a plurality of training samples including a plurality of sets of syndrome bits into which noise has been injected.

[0019] In a further implementation of the first, second and / or third aspect, the quantum error correcting code is one of the group consisting of a stabilizer code, a surface code, and a topological code.

[0020] In a further implementation of the first, second and / or third aspects, the loss function of the LER is defined based on a binary cross-entropy loss over the predicted soft errors calculated by the trained Transformer neural network-based decoder.

[0021] In one optional implementation of the first, second and / or third aspects, the loss function of the LER is redefined as a differentiable loss function using a differentiable equivalent mapping of a binary mod 2 based XOR operation, thereby enabling minimization of the differentiable loss function of the LER.

[0022] In one optional implementation of the first, second and / or third aspects, the differentiable loss function of the LER is redefined based on a binarization of the predicted soft errors.

[0023] In one optional implementation of the first, second and / or third aspects, the differentiable loss function of the LER is regularized.

[0024] In a further implementation of the first, second, and / or third aspects, the one or more noise estimators include one or more shallow neural networks trained to calculate the initial noise estimates by minimizing binary cross-entropy loss between syndrome bits using a plurality of training samples including syndrome bits of the one or more codewords encoded using the one or more quantum error correcting codes.

[0025] In a further implementation of the first, second and / or third aspects, the one or more encoded codewords used to create the training samples include a zero codeword.

[0026] Other systems, methods, features, and advantages of the present disclosure will be or become apparent to one with skill in the art upon examination of the following figures and detailed description. It is intended that all additional systems, methods, features, and advantages be included within the following description, be within the scope of the present disclosure, and be protected by the accompanying claims.

[0027] Unless otherwise defined, all technical and / or scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which this invention belongs. Although methods and materials similar or equivalent to those described herein can be used to practice or test embodiments of the present invention, exemplary methods and / or materials are described below. In case of conflict, the patent specification, including definitions, will prevail. In addition, the materials, methods, and examples are merely illustrative and are not necessarily intended to be limiting.

[0028] Implementation of methods and / or systems according to embodiments of the present invention may include automatically performing or completing selected tasks. Furthermore, depending on the actual equipment and devices in an embodiment of a method and / or system of the present invention, some selected tasks may be implemented by hardware, software, or firmware, or a combination thereof using an operating system.

[0029] For example, hardware for performing selected tasks according to embodiments of the present invention may be implemented as a chip or circuit. Selected tasks according to embodiments of the present invention may be implemented as software, as a plurality of software instructions executed by a computer using any suitable operating system. In an example embodiment of the present invention, one or more tasks according to an example embodiment of the methods and / or systems described herein are performed by a data processor, such as a computing platform for executing a plurality of instructions. Optionally, the data processor includes volatile memory for storing instructions and / or data, and / or non-volatile storage, e.g., a magnetic hard disk and / or removable media, for storing instructions and / or data. Optionally, a network connection is also provided. A display device and / or a user input device (e.g., a keyboard or mouse) are further provided, as needed. [Brief explanation of the drawings]

[0030] Some embodiments of the present invention are described herein, by way of example only, with reference to the accompanying drawings. Referring now in detail to the following drawings, it is emphasized that these details are given by way of example only and are for the purpose of illustratively discussing embodiments of the present invention. Similarly, from viewing the description together with the drawings, it will become apparent to those skilled in the art how embodiments of the present invention may be practiced. [Figure 1]FIG. 1 is a schematic diagram of an exemplary transmission system comprising a neural network-based decoder trained to decode quantum error correcting codes transmitted over a transmission channel, in accordance with some embodiments of the present invention. [Figure 2] FIG. 1 is a schematic diagram of an exemplary Transformer Neural Network-based decoder trained to decode an error-correcting code transmitted over a transmission channel. [Figure 3] FIG. 1 is a schematic diagram of an exemplary Transformer Neural Network-based decoder trained to decode a quantum error correcting code transmitted over a transmission channel, according to some embodiments of the present invention. [Figure 4A] FIG. 10 is a schematic diagram of exemplary masks calculated for a Transformer Neural Network based decoder based on a graphical representation of each error correcting code, in accordance with some embodiments of the present invention. [Figure 4B] FIG. 10 is a schematic diagram of exemplary masks calculated for a Transformer Neural Network based decoder based on a graphical representation of each error correcting code, in accordance with some embodiments of the present invention. [Figure 5] 1 is a flowchart of an exemplary process for training a Transformer Neural Network-based decoder for decoding quantum error-correcting codes according to some embodiments of the present invention. [Figure 6] 1 is a flowchart of an exemplary process for decoding a quantum error-correcting code using a decoder based on a trained Transformer neural network, according to some embodiments of the present invention. [Figure 7] FIG. 2 is a schematic diagram of a lattice representation of an exemplary toric code used by a decoder based on a Transformer Neural Network, according to some embodiments of the present invention; [Figure 8A]1 is a graphical chart comparing decoding performance for several legacy decoders and a decoder based on a Transformer Neural Network applied to decoding a quantum error correcting code, according to some embodiments of the present invention. [Figure 8B] 1 is a graphical chart comparing decoding performance for several legacy decoders and a decoder based on a Transformer Neural Network applied to decoding a quantum error correcting code, according to some embodiments of the present invention. [Figure 8C] 1 is a graphical chart comparing decoding performance for several legacy decoders and a decoder based on a Transformer Neural Network applied to decoding a quantum error correcting code, according to some embodiments of the present invention. [Figure 8D] 1 is a graphical chart comparing decoding performance for several legacy decoders and a decoder based on a Transformer Neural Network applied to decoding a quantum error correcting code, according to some embodiments of the present invention. [Figure 8E] 1 is a graphical chart comparing decoding performance for several legacy decoders and a decoder based on a Transformer Neural Network applied to decoding a quantum error correcting code, according to some embodiments of the present invention. [Figure 8F] 1 is a graphical chart comparing decoding performance for several legacy decoders and a decoder based on a Transformer Neural Network applied to decoding a quantum error correcting code, according to some embodiments of the present invention. [Figure 9A] 1 is a graphical chart illustrating the effect of design parameters on the decoding performance of a decoder based on a Transformer Neural Network applied to decoding quantum error correcting codes according to some embodiments of the present invention. [Figure 9B]1 is a graphical chart illustrating the effect of design parameters on the decoding performance of a decoder based on a Transformer Neural Network applied to decoding quantum error correcting codes according to some embodiments of the present invention. [Figure 9C] 1 is a graphical chart illustrating the effect of design parameters on the decoding performance of a decoder based on a Transformer Neural Network applied to decoding quantum error correcting codes according to some embodiments of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0031] Some embodiments of the present invention relate to training and using neural networks to decode quantum error-correcting code words transmitted over transmission channels susceptible to interference, and more particularly to training and using decoders based on Transformer neural networks to quantum-decode error-correcting code words transmitted over transmission channels susceptible to interference.

[0032] Wired and / or wireless transmission channels are fundamental building blocks for data transmission applications, such as, for example, communication channels, network links, memory interfaces, inter-component connections (buses, switched fabrics, etc.), and / or the like. However, data transmitted over transmission channels that are subject to one or more interferences, such as, for example, noise, crosstalk, attenuation, and / or the like, can often suffer from interference-induced errors.

[0033] Thus, error-correcting codes may be applied at the physical communication layer to encode codewords transmitted over a transmission channel, thereby enabling a receiving decoder to efficiently detect and possibly correct errors in the transmitted encoded codewords, thereby improving the decoder's efficiency in accurately recovering the codewords while maintaining a high transmission rate.

[0034] Quantum computing may experience similar transmission channel interference, presenting additional challenges not present in classical (non-quantum) computing and communications domains. First, the no-cloning theorem for quantum states prevents cloning of quantum states and therefore the addition of arbitrarily redundant parity information, as is done in classical ECC. Furthermore, data in the quantum domain can suffer from phase inversions in addition to bit inversions, whereas classical ECC only deals with bit-flip errors. Also, while bit measurements are standard in classical ECC, the phenomenon of wave function collapse precludes direct measurement of quantum data qubits because such measurements would collapse the wave function and erase the encoded quantum information.

[0035] These challenges may prevent the trivial application of existing ECC techniques in the quantum computing domain.

[0036] According to some embodiments of the present invention, methods and systems are provided for constructing novel neural network-based decoder architectures for decoding quantum error correcting codes (QECC) transmitted over interference-prone transmission channels.

[0037] In particular, the neural network based decoder architecture may rely on, for example, a Transformer Neural Network with a self-attention decoding layer specifically adapted to the quantum error correcting code used to encode the codeword, e.g., stabilizer codes, topological codes, surface codes, etc.

[0038] In this specification, the Transformer Neural Network Based QECC Decoder (hereinafter also referred to as QECCT) utilizes state-of-the-art Transformer Neural Network Based ECC Decoder (ECCT) technology with some modifications that adapt it for efficient and high performance QECC decoding.

[0039] Thus, similar to an ECCT decoder, a QECCT decoder may comprise an input layer adapted to create an embedding of a received encoded codeword, multiple self-attention decoding layers adapted and trained for decoding, and an output layer adapted to reduce the dimensionality of the concatenated decoded output from the decoding layers and recover the encoded codeword.

[0040] However, in contrast to the ECCT decoder, the magnitude of the codeword qubits cannot be measured, so the QECCT decoder, and specifically its input layer, is adapted to create a high-dimensional embedding for the initial noise estimates calculated for the syndrome bits of the codeword that can be measured.

[0041] An initial noise estimate may be calculated for the syndrome bits using one or more noise estimators, optionally implemented using one or more trained neural networks, typically shallow neural networks. Due to the possibility of measurement errors, the syndrome bits may be repeatedly sampled to overcome measurement error limitations.

[0042] Also, rather than recovering the encoded codeword, as is done by an ECCT decoder, the decoding layer of the QECCT decoder is adapted to recover a logical operator mapping of the encoded codeword that commutes with the encoded codeword. In particular, the decoding layer of the QECCT decoder may recover a matrix of logical operators according to a mask derived from a parity check matrix that indicates the relationship between each pair of syndrome bits based on the embedding created for the syndrome bits. Thus, the QECCT decoder may be conditioned to decode the logical operator mapping of the codeword based on the connected syndrome bits while ignoring the unconnected syndromes masked by the mask.

[0043] During training of the QECCT decoder, the decoding layer is trained to optimize a joint loss function (objective) that combines the loss on the logical operators (called the Logical Error Rate (LER)), the loss on the data (called the Bit Error Rate (BER)), and the loss on the noise estimator that is applied to calculate the initial noise estimate.

[0044] Because the LER loss function (objective function) can be highly non-differentiable and therefore difficult to optimize, the LER loss function can be redefined in an equivalent form that makes the non-differentiable LER loss function differentiable. The differentiable LER loss function can be manipulated to apply binary quantization and regularization.

[0045] QECCT decoders may offer significant advantages and benefits over existing quantum error correcting code decoders, including existing quantum error correcting code decoders that employ neural network decoding.

[0046] First, by adapting a QECC decoder to decode the logical operator mapping of codewords encoded using QECC, the inherent limitations of the quantum computing domain discussed above can be overcome: unclonability, bit and phase flips, and wave function collapse.

[0047] Furthermore, due to the sparse design of the quantum error correcting codes on which the QECCT decoder is conditioned, training and deploying the QECCT decoder can be very simple and affordable in terms of computational resources and / or time compared to existing neural network decoders, both model-free and model-based.

[0048] Furthermore, by masking the non-connected syndrome bits and making the QECCT decoder rely only on the connected bits, the decoding performance of the QECCT decoder, such as accuracy, reliability, consistency, robustness, etc., can be significantly improved, and the decoding calculation resources, such as processing resources, memory resources, calculation time, etc., can be significantly reduced.

[0049] Before describing at least one embodiment of the invention in detail, it is to be understood that the invention is not necessarily limited in its application to the details of construction and the arrangement of elements and / or methods set forth in the following description and / or illustrated in the drawings and / or examples. The invention is capable of other embodiments and of being practiced or carried out in various ways.

[0050] Those skilled in the art will appreciate that aspects of the present invention may be embodied as a system, method, or computer program product. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, microcode, etc.), or an embodiment combining software and hardware aspects, all of which may be collectively referred to as a "circuit," "module," or "system." Furthermore, aspects of the present invention may take the form of a computer program product embodied in one or more computer-readable medium(s) having computer-readable program code stored thereon.

[0051] Any combination of one or more computer-readable media may be utilized. A computer-readable storage medium may be a tangible device capable of retaining and storing instructions for use by an instruction execution device. The computer-readable storage medium may be, for example, but is not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination thereof. A non-exclusive list of more specific examples of computer-readable storage media includes portable computer diskettes, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random access memory (SRAM), portable compact disc read-only memory (CD-ROM), digital versatile disc (DVD), memory stick, floppy disk, mechanically encoded devices such as punch cards or ridge structures in grooves with instructions recorded thereon, and any suitable combination of the above. As used herein, a computer-readable storage medium should not be considered to be a transitory signal itself, such as an electric wave, an electromagnetic wave through a transmission medium such as a waveguide (e.g., light pulses through a fiber optic cable), or an electrical signal through a wire, etc.

[0052] Computer program code including computer readable program instructions embodied in a computer readable medium may be transmitted using any suitable medium, including, but not limited to, wireless, wireline, fiber optic cable, RF, etc., or any suitable combination thereof.

[0053] The computer-readable program instructions described herein can be downloaded from a computer-readable storage medium to each computing / processing device or to an external computer or external storage device, for example, via the Internet, a local area network (LAN), a wide area network (WAN), and / or a wireless network. The network may include copper transmission cables, fiber optic transmissions, wireless transmissions, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface in each computing / processing device receives the computer-readable program instructions from the network and forwards the computer-readable program instructions for storage in a computer-readable storage medium in each computing / processing device.

[0054] Computer readable program instructions for carrying out operations of the present invention may be written, for example, as assembler instructions, Instruction-Set-Architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, state-setting data, or as either source code or object code written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Smalltalk, C++, and the like, as well as conventional procedural programming languages ​​such as the C programming language or similar programming languages.

[0055] The computer-readable program instructions can execute entirely on the user's computer, partially on the user's computer, as a stand-alone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In this case, the remote computer may be connected to the user's computer via any type of network, such as a local area network (LAN) or wide area network (WAN), or may be connected to an external computer (e.g., using the Internet via an Internet Service Provider (ISP)). In some embodiments, electronic circuitry, including, for example, programmable logic circuits, field-programmable gate arrays (FPGAs), or programmable logic arrays (PLAs), may execute the computer-readable program instructions by utilizing state information from the computer-readable program instructions to personalize the electronic circuitry to perform aspects of the present invention.

[0056] One aspect of the present invention is described with reference to flowcharts and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowcharts and / or block diagrams, and combinations of blocks in the flowcharts and / or block diagrams, can be implemented by computer-readable program instructions.

[0057] The flowcharts and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block in the flowcharts or block diagrams may represent a module, segment, or portion of instructions, which includes one or more executable instructions for implementing the specified logical function(s). In some alternative embodiments, the functions shown in the blocks may be performed in an order different from that shown in the figures. For example, two blocks shown in succession may in fact be executed substantially concurrently, or the blocks may be executed in the reverse order, depending on the functionality involved. It should also be noted that each block of the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, may be implemented by a specific hardware-based system that performs the specified functions or operations or executes a combination of specific hardware and computer instructions.

[0058] Referring to the drawings, FIG. 1 is a schematic diagram of an exemplary transmission system comprising a neural network-based decoder trained to decode a quantum error correcting code transmitted over a transmission channel, in accordance with some embodiments of the present invention.

[0059] As an example, the transmission system 100 may include an encoder 102 adapted to encode data (messages), in particular quantum data, to be transmitted over a transmission channel 106 that can be decoded by a decoder 104 adapted to decode and recover the encoded data.

[0060] The encoder 102 and decoder 104 are part of a transmitter and receiver, respectively (not shown), which may include one or more additional circuits, modules, and / or functions. For example, in the case of a communications system, the transmitter may include a modulator configured to modulate the quantum data encoded by the encoder 102 according to one or more modulation schemes known in the art.

[0061] The transmission channel 106, which may include one or more wired and / or wireless transmission channels, may be intended for one or more of a number of applications, such as, for example, a communications channel, a network link, a memory interface, an inter-component connection (e.g., a bus, a switched fabric, etc.).

[0062] In particular, the transmission channel 106 may be subject to, for example, noise, crosstalk, attenuation, and / or one or more interferences that may induce one or more errors in the transmitted data.

[0063] Therefore, to overcome data corruption caused by interference in quantum systems, the encoder 102 is configured to encode the transmitted quantum data according to one or more quantum error correcting codes (QECCs), models, and / or protocols known in the art to support error detection and / or correction, such as, for example, stabilizer codes, topological codes, surface codes, and / or the like.

[0064] The decoder 104 may be a neural network-based decoder 204, which typically comprises one or more neural networks trained to decode QECC codes, such as deep learning (DL) neural networks, e.g., fully connected (FC) neural networks, convolutional neural networks (CNN), feed-forward (FF) neural networks, recurring neural networks (RNN), etc.

[0065] In particular, similar to the ECCT, the decoder 104 may employ a model-free transformer architecture, which means that the neural network-based decoder 104 does not depend on a specific decoding model, such as Belief Propagation (BP). Therefore, the transformer neural network-based QECC decoder 104 is also interchangeably referred to as a QECC transformer (QECCT).

[0066] Adapting existing classical ECC methods to QECC is not easy, as the inherent mechanisms and principles of quantum computing may pose challenges not experienced in traditional information technology, particularly with regard to error-correcting codes (ECC).

[0067] The first difficulty in applying ECC-based knowledge to QECC stems from the quantum state cloning theorem, which asserts that quantum states cannot be cloned and therefore arbitrarily redundant parity information cannot be added, as is done in classical ECC. A second challenge is the need to detect and correct quantum successive bit and phase flips, whereas classical ECC only targets bit-flip errors. A third major challenge is the wave function collapse phenomenon, which, although standard in ECC, prevents direct measurement of qubits because it causes wave function collapse and erases the encoded quantum information.

[0068] To overcome these challenges, QECC schemes and methods have been developed. For example, the threshold theorem shows that increasing the distance of a code results in a corresponding reduction in the logical error rate, meaning that quantum error-correcting codes can arbitrarily suppress the logical error rate. This increase in distance can be obtained by developing encoding schemes that reliably store and process information in a set of logical qubits by redundantly encoding the information onto a larger set of less reliable physical qubits.

[0069] Most current QECC methods belong to the category of stabilizer codes, which can be considered a generalization of classical linear codes. Similar to classical parity check constraints, a group of stabilizer operators may provide syndromes that preserve the logical quantum state and allow error detection.

[0070] Optimal decoding can be defined by an infeasible NP-hard maximum likelihood rule. To support efficient decoding, considerable research has been done on the design of codes with some additional algebraic structure, for example, surface codes, which are derived from topological codes, especially toric codes.

[0071] Topological QECC codes may encode each logical qubit in a two-dimensional (2D) lattice of physical qubits. This local design of the code via nearest-neighbor coupled qubits allows for the correction of a wide range of errors. Furthermore, under certain assumptions, surface codes can result in an exponential reduction in error rate.

[0072] Decoder 104 may employ one or more Transformer Neural Network based decoders, as known in the art, to decode the ECC codes.

[0073] FIG. 2 is a schematic diagram of an exemplary Transformer neural network-based decoder trained to decode an error-correcting code transmitted over a transmission channel.

[0074] As an example, transmission system 200 may include a transmitter 210 that may transmit data (messages) to a receiver 212 via a transmission channel 206, such as transmission channel 106, which may be subject to one or more interferences. Transmitter 102 may include an encoder 202 configured to encode the transmitted data according to one or more ECC codes, models, and / or protocols, such as, for example, algebraic linear codes, polar codes, LDPC, HDPC, and / or similar linear block codes. However, the ECC codes may also include non-block codes, such as, for example, convolutional codes, and may also include non-linear codes, such as, for example, Hadamard codes.

[0075] The transmitter 210, which is shown in general terms only, may further include one or more additional circuits, modules, and / or functions, such as, for example, a modulator 208 adapted to modulate the data encoded by the encoder 202 according to one or more modulation schemes, such as, for example, binary phase-shift keying (BPSK), quadrature phase shift keying (QPSK), and / or the like.

[0076] Also, the receiver 212, which is referred to in general terms only, may include a neural network-based decoder 204, including, among other things, one or more trained neural networks known as DL neural networks, Transformer neural networks, FC neural networks, CNNs, FF neural networks, RNNs, etc. In particular, the decoder 204 employs a model-free transformer architecture and may be interchangeably referred to as an ECC transformer (ECCT).

[0077] A classical ECC linear code C is a binary Galois field GF(2), i.e., a Galois field of order 2 over which GH T = 0, the matrix may be defined by a binary generator matrix G of size k × n and a binary parity check matrix H of size (nk) × n.

[0078] Thus, the encoder 202 applying the generator matrix G receives the input data message m∈{0,1} k Let x∈C⊂{0,1} n A codeword x=G of size k×n() satisfies Hx=0. T m∈{0,1} n may be encoded as

[0079] The modulator 208 modulates the coded codeword x according to one or more modulation schemes (e.g., BPSK), i.e., over {±1}, to generate x representing the modulation of x. s Generate the modulated encoded codeword x s may be transmitted over a transmission channel 206, for example, a symmetric (potentially binary) transmission channel, for example, an Additive White Gaussian Noise (AWGN) channel.

[0080] The output of the transmit channel 206 received by the receiver 212 is

number

[0081] Decoder 204 (f:S→{0,1} n ) goal is to

number

[0082]

number

number

[0083] When discussing quantum error correcting codes (QECC), the fundamental transition to the quantum realm is defined by the transition from classical bits to quantum bits (qubits), and the quantum state of the qubits

number

number

[0084] A coherent quantum error process E can be decomposed into a sum of operators from the Pauli set {I, X, Z, XZ}, where the Pauli basis is defined by the identity map I, the qubit flip X, and the phase flip Z, as expressed by Equation 3 below.

number

[0085] Thus, the single qubit error may be defined by Equation 4 below:

number

number

[0086] According to the no-cloning theorem, the quantum state

number

number

number

number

number

number

[0087] An example of such a single operator is the GHZ state (Greenberger, Horne, and Zeilinger) produced by a CNOT gate.

[0088]

number

number

[0089] The orthogonality between C and F allows the subspace occupied by a logical qubit to be determined by projective measurements without compromising the encoded quantum information. In the context of quantum encoding, this type of non-destructive measurement set P is called stabilizer measurements and can be performed via additional qubits, e.g., ancillary bits.

[0090] The results of the stabilizer measurements for a given state are called syndromes, and are given stabilizer generators.

number

number

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number

[0091] An important class of Pauli operators is the class of logical operators. These operators are not elements of the stabilizer group, but may commute with all stabilizers. The stabilizer operators are located in the sign space, i.e.,

number

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[0092] Such operators may commute with stabilizers, but may also represent undetectable errors. Thus, similar to classical information bits, QECC benchmarks generally employ logical error metrics, which are the sum of the predicted projection noise.

number

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[0093] Referring again to FIG. When dealing with QECC in terms of ECC, another way to represent the stabilizer code is to write the block parity check matrix as

number

[0094] Quantum decoder 104(f{0,1} |P| →{0,1} |L| The main purpose of ) is to provide an approximation of the noise given only the syndrome s.

[0095] Therefore, a quantum data decoding scheme can be reduced to its classical counterpart as follows: k logical qubits are analogous to k classical information bits, and n physical qubits are analogous to a classical codeword. The syndrome of a quantum state can be calculated or simulated in a classical manner by defining a binary parity check matrix constructed based on the code quantum stabilizer.

[0096] However, the decoder 102 adapted for QECC decoding has some differences compared to classical ECC: first, there is no access to the current state, and no arbitrary measurement of y is possible, as is standard for classical ECC; also, the objective is logical qubits, and the code is a logical operator mapping

number

[0097] The goal of the decoder 104 is to

number

[0098] The decoder 104 may be built based on an ECC transformer architecture with some modifications.

[0099] The input h(y) to the Transformer Neural Network QECC Decoder (QECCT) may be defined by a concatenation of the codeword-independent magnitude and syndrome s,

number

number

[0100] Each element can then be embedded in a high-dimensional space to make it more expressive, for example, the initial position embedding Φ is

number

number

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[0101] The interaction between bits is naturally achieved through the self-attention module of the QECCT decoder 104 combined with a binary mask derived from the parity check matrix expressed as Equation 5 below.

[0102]

number

[0103] Masking allows incorporating sparse and efficient information about the code while avoiding the loop vulnerability of belief propagation-based decoders.

[0104] Finally, the transformed embedding is projected into a 1-dimensional vector for noise prediction. The computational complexity is O(N(d 2 n+hd), where N is the number of layers of the QECCT decoder 104, n is the code length, and h<<n 2 denotes the number of elements in the mask, which can typically be very small for toric codes, surface codes, and / or the sparse codes they contain.

[0105] Please refer to FIG. 3, which is a schematic diagram of an exemplary Transformer Neural Network based decoder trained to decode a quantum error correcting code transmitted over a transmission channel.

[0106] For example, a Transformer Neural Network-based decoder 104A, such as a Transformer Neural Network-based decoder 104 (QECCT) adapted and trained to decode one or more codewords encoded using a QECC, such as a stabilizer code, a toric code, a topological code, a surface code, etc., transmitted over an interference-subjected transmission channel, such as transmission channel 106, may be comprised of an input layer 302, multiple (N) decoding layers 304, and an output layer 306.

[0107] As above, the binary parity check matrix of QECC is denoted by H, noise, e.g., binary noise, is denoted by ε, the syndrome (bit) is denoted by s = Hε, and the binary matrix of logical operators is

number

[0108] Furthermore, as described herein, the parity check matrix H may comprise a bit-flip parity check matrix calculated to correct bit flips of the quantum bits, and a phase-flip parity check matrix calculated to correct phase flips of the quantum bits.

[0109] The input layer 302 receives the syndrome bits s of the syndrome s of the received encoded codeword. i A noise estimation unit g estimates the noise injected into ω The initial noise estimate {s i} i=1 T and can be adapted to create an embedding for the syndrome bits according to an initial noise estimate.

[0110] Syndrome decoding is a well-known procedure in ECC, but most common decoders, especially those based on neural networks, assume the availability of arbitrary measurements of the output of the transmission channel 106. However, in the QECC setting, only the syndrome s is available, since classical measurements are not allowed due to the wave function collapse phenomenon.

[0111] Therefore, to overcome measurement collapse, the classical ECC Transformer Neural Network based decoder (ECCT) can be extended to a QECC Transformer Neural Network based decoder (QECCT) by replacing the magnitude of the channel output y with an initial estimate of the noise in syndrome s, which is further refined by a code recognition network. In other words, the channel output magnitude measurement h(y) = [|y|, 1-2s] is calculated by h q (s)=[gω(s),s].

[0112] Noise Estimation Unit

number

[0113] As a nonlinear transformation of syndrome s, the noise estimation part g ω Since is independent of the quantum state / codeword, it can be highly robust against overfitting. ωmay be trained using the loss function (objective) expressed in Equation 6 below:

number

[0114] The shift from using the magnitude of the output of the transmission channel to using an initial error estimate is important for overcoming quantum measurement collapse and, as will be explained below, can significantly improve the decoding performance of decoder 104A.

[0115] The embedding created by the input layer 302 may obviously have a higher dimension than the dimension of the received initial noise estimate. For example, for a code length of n, the embedding may be of dimension n+s.

[0116] In contrast to classical ECC, quantum error correction aims to restore noise to the logical origin of the code, so that several solutions may be valid error corrections.

[0117] ECCT f θ The output of decoder 104A is

number

number

[0118] Therefore, the decoding layer 304 may be adapted to calculate an estimated logical operator matrix for the received encoded codeword. Each of the multiple decoding layers 304 may have a self-attention layer with one or more heads constructed according to a mask indicating the relationship between the embeddings. The mask indicating the relationship between the embeddings is derived from the parity check matrix H of the quantum error correcting code. Therefore, the mask may be adapted to unmask pairs of connected parity bits and mask pairs of disconnected parity bits.

[0119] In particular, ECCT f θ may process the estimated noise input and perform decoding by analyzing the input syndrome interactions according to a mask obtained, derived, and / or calculated according to the QECC used to encode the received codeword. The mask is a mask representing which pairs of bits s in syndrome s have high correlation and which pairs of bits do not have high correlation. i It may also show the relationship between

[0120] The mask may be created, for example, based on an extended bipartite graph representation of the parity check matrix H of the quantum error correcting code, such as a Tanner graph, a factor graph, and / or other graphs including multiple nodes connected via multiple edges, where each pair of connected bits includes bits that share one or more of the multiple nodes, and each pair of unconnected bits includes bits that do not share any of the multiple nodes.

[0121] Using the mask, the decoder 104A, specifically the self-attention mechanism of the decoding layer 304, can take into account bits that are related to each other with respect to the QECC, i.e., for example, the parity check matrix H, which is a stabilizer. By using adaptive masking derived from the parity check matrix H, the Transformer Neural Network-based decoder 104A (QECCT) can learn the dependencies between related qubits. However, it is important to note that similar extensions are not limited to Transformer Neural Networks and can easily be applied to other neural decoder architectures.

[0122] Reference is made to FIGS. 4A and 4B, which show schematic diagrams of exemplary masks calculated for a Transformer Neural Network based decoder based on a graphical representation of each error correcting code, according to some embodiments of the present invention.

[0123] Diagrams 400 and 410 provide parity check matrices for two-toric and four-toric codes, respectively. Diagrams 402 and 412 show masks derived for two-toric and four-toric codes from the respective parity check matrices. Each parity check matrix comprises a first block matrix for bit reversal X and a second block matrix for phase reversal Z.

[0124] As can be seen, toric codes are characterized by high sparsity induced by a code architecture that exhibits high locality: bits that are close to each other may have high correlation (connectivity), while bits that are far from each other may have low correlation (disconnectivity). This locality is expressed in a mask that may reflect only stabilizer-related elements.

[0125] Again, refer to FIG. Since the decoding layer 304 is adapted to calculate an estimated logical operator matrix of the received encoded codeword, the metric used to train and optimize the decoding layer 304 may be the logical error rate (LER), which may provide valuable information about the actual decoding performance.

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[0126] Additionally, the loss function (objective function) used to train and optimize the decoding layer 304 may employ a differentiable equivalent mapping of XOR, i.e., sum in GF(2). Defining the bipolar mapping φ:{0,1}→{±1} on GF(2) as φ(u)=1−2u,u∈{0,1}, the property

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[0127] Thus, differentiable functions

number

number

[0128] As shown in the above equation, the LER loss function used to train the decoding layer 304 may be defined based on the binary cross-entropy loss over the predicted soft errors calculated by the trained Transformer neural network-based decoder 104A. Furthermore, the LER loss function may be redefined as a differentiable loss function using a differentiable equivalent mapping of the XOR operation based on binary mod 2 (GF(2)), which enables minimization of the LER differentiable loss function. In particular, the differentiable LER loss function may be defined based on the binarization of the predicted soft errors of the logical operator matrix of the codewords encoded using the QECC code.

[0129] The binary quantization of activations in the decoding layer 304 is performed using the sigmoid function bin(x)=σ(x)=(1+e-x ) -1 This may be based on a differentiable approximation by, which may improve decoding performance compared to other binary quantization methods known in the art, such as, for example, a Straight-Through Estimator (STE).

[0130] In addition to directly minimizing the LER metric, e.g., L BER =BCE(f θ It is desirable to identify a noise prediction solution that is close to the actual system noise by regularizing the differentiable loss function (objective) of LER with the classical bit error rate (BER) loss function (objective) defined by (s), ε).

[0131] Therefore, the overall loss function (objective) for training the decoding layer 304 aims to minimize the Linear Error Rate (LER), Bit Error Rate (BER), and Noise Estimator Error Rate, as expressed in Equation 11 below: L=λ BER L BER +λ LER L LER +λ g L g (11) where λ BER , λ LER , and λ g are the loss functions L BER , L LER , and L g indicates the weight of

[0132] In the presence of measurement error, each syndrome measurement may be repeated T times, thereby augmenting the input to decoder 104A with an additional time dimension. Formally, if the binary system noise is {ε t} t=1 T and the binary measurement noise is

number

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[0133] To be invariant to the number of measurements, each measurement is first analyzed separately, and then global decoding is performed by applying, for example, a symmetric average pooling function in the middle of the neural decoder 104A. The NN decoder 104A has N decoding layers 304 and a hidden activation tensor ℓ at layer l=N / 2.

number

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[0134] Therefore, the loss function L g can be defined as the distance between the pooled embedding and the noise, as expressed in Equation 13 below:

number

[0135] As the ECCT architecture is extended to its quantum counterpart (QECCT), the hidden activation tensor has the shape

number

[0136] The initial encoding performed in the input layer 302 is performed by setting the number of physical qubits to n and the length of the syndrome to n. s Then, n+n s It can be defined as a d-dimensional one-hot encoding of the input elements. ω may, for example, comprise two Fully Connected (FC) layers with hidden dimension equal to 5n and GELU nonlinearity.

[0137] Decoding may be performed by a concatenation of N decoding layers 304, each containing a self-attention layer and a feedforward layer, with normalization layers inserted between them. For fault syndrome measurements, the [N / 2]th layer may perform average pooling over the time dimension.

[0138] The output layer 306 is adapted to generate a vector representing the predicted soft errors of the logical operator matrix of the received codeword, and may be configured to reduce the dimension of the soft error vector concatenating the multiple soft error vectors calculated by the multiple decoding layers 304 based on the embedding created by the input layer 302.

[0139] For example, the output layer 306 may comprise two fully connected (FC) layers, the first of which converts the element-wise embedding into a one-dimensional n+n s vector, and the second layer is configured to further reduce the vector to an n-dimensional vector representing soft-decoded noise trained over the loss function (objective) of Equation 11.

[0140] The logical operator L can be applied to the estimated soft errors in the logical operator matrix h(L, σ(z)) to decode the encoded codeword z output from decoder 104A.

[0141] The complexity of the decoder 104A is linear in the code length n and quadratic in the embedding dimension d, e.g., for topological codes, it is O(Nd 2 n).

[0142] As previously explained, the decoder 104A, and specifically the decoding layer 304, may be trained by minimizing and / or optimizing the loss function (objective) of Equation 11.

[0143] FIG. 5 is a flowchart of an exemplary process for training a Transformer Neural Network-based decoder for decoding quantum error correcting codes according to some embodiments of the present invention.

[0144] For example, process 500 may be performed to train a decoder 104 (e.g., a QECC decoder based on one or more Transformer neural networks, such as decoder 104A).

[0145] Process 500 may be performed by one or more systems, platforms, and / or services hereinafter collectively referred to as a training system. The training system comprises one or more hardware processors adapted to execute program instructions stored in non-transitory storage (program storage). The hardware processors that optionally perform training process 500 may be supported by one or more hardware elements available to and / or utilized by the training system, such as, for example, an artificial intelligence (AI) accelerator, a graphics processing unit (GPU), and / or the like.

[0146] Optionally, the training system may include one or more network interfaces for connecting to one or more wired and / or wireless networks, such as a local area network (LAN), a wireless LAN (WLAN, e.g., Wi-Fi), a wide area network (WAN), a municipal area network (MAN), a cellular network, the Internet, and / or the like. Via the network, the training system may communicate with one or more remote resources, such as, for example, servers, cloud services, databases, storage resources, etc.

[0147] Process 500 begins with a training system obtaining a plurality of training samples, as shown at 502. The training system may obtain the training samples, for example, by fetching, retrieving, receiving, etc., the training samples from one or more sources, such as a local storage device (e.g., a hard drive, memory, etc.), a remote resource accessible over a network (e.g., a database, storage server, etc.), etc.

[0148] The training samples are then coded using one or more quantum error correction codes (QECCs), and one or more noise estimators (e.g., noise estimators g) are used to estimate noise injected into syndrome bits of one or more codewords transmitted over one or more transmission channels, such as interference-affected transmission channel 106. ω The noise estimation unit may include multiple initial noise estimates calculated by a neural network based noise estimator (such as a noise estimation unit based on a neural network).

[0149] Optionally, the training samples may include an initial noise estimate calculated for a zero codeword encoded with a QECC code and transmitted over a transmission channel 106 subject to interference.

[0150] As shown at 504, a training system may use a plurality of training samples to train a Transformer Neural Network based decoder 104A to decode codewords encoded using a QECC code.

[0151] In particular, as previously described herein, the decoder 104A includes: (1) an LER, BER, and noise estimator g ω It can be trained to decode QECC-encoded codewords by (1) calculating an estimated logical operator matrix for the training codeword to minimize a joint loss function (objective) whose objective is to minimize the error rate of .

[0152] The training system may output a trained Transformer neural network-based decoder 104A for decoding one or more codewords, particularly previously unseen codewords, encoded using one or more QECC codes, as shown at 506. For example, the trained decoder 104A may be used by one or more decoding systems adapted to decode codewords encoded using one or more QECC codes.

[0153] The training data set may be defined, constructed, and / or selected depending on, for example, the architecture of the decoder 104A, the desired decoding performance, the capacity and / or computational resources of the training system (e.g., processing resources, storage resources, computation time, etc.), and / or one or more other parameters.

[0154] For example, an exemplary training session for an exemplary decoder 104A with N=6, i.e., six decoding layers 304, and d=128, may be conducted based on a training data set containing 512 training samples per mini-batch for 200-800 epochs depending on the QECC code length, with 5000 mini-batches per epoch.

[0155] Training may be done by randomly sampling noise within the physical error rate test range. The default weight parameters chosen for the loss function (objective) are λ BER =0.5, λ LER = 1, and λ g =0.5. Note that other configurations and / or longer training data sets may be beneficial and improve the decoding performance of decoder 104A.

[0156] The learning rate of the decoder 104A is, for example, 5·10 at the end of training. -7 5·10 with cosine decay scheduler up to -4 The warm-up known in the prior art was not employed.

[0157] Training times can range from 153 to 692 seconds per epoch for code lengths of 32 to 400, respectively, for decoder 104A with the default N=6 and d=128 architecture.

[0158] Reference is now made to FIG. 6, which is a flowchart of an exemplary process for decoding a quantum error correcting code using a decoder based on a trained Transformer neural network, according to some embodiments of the present invention.

[0159] For example, process 600 may be performed using decoder 104A (e.g., a QECC decoder based on a Transformer neural network, such as decoder 104) to decode one or more codewords encoded using one or more QECC codes.

[0160] Process 600 may be performed by one or more systems, platforms, and / or services (collectively referred to hereinafter as decoding systems) that include one or more hardware processors adapted to execute program instructions stored in non-transitory storage (program storage). The hardware processors that optionally execute decoding process 600 may be supported by one or more hardware elements available to and / or utilized by the decoding system, such as, for example, AI accelerators, GPUs, etc.

[0161] As shown at 602, process 600 includes estimating noise g of noise injected into syndrome bits of one or more received codewords encoded using a quantum error correction code and transmitted over a transmission channel, such as transmission channel 106, subject to interference. ω The decoding system may start by receiving an initial noise estimate calculated by one or more noise estimators, such as:

[0162] As shown in 604, the decoding system applies a trained neural network based decoder 104A, such as trained decoder 104, adapted and trained to calculate an estimated logical operator matrix for the received codeword and generate a vector representing predicted soft errors in the logical operator matrix.

[0163] As shown at 606, the decoding system may output a vector representing the predicted soft errors.

[0164] While decoder 104 may be general-purpose with respect to the QECC code it decodes, decoder 104A, for example, is more applicable and efficient to topological codes, e.g., a mixture of surface codes and toric codes (variants of surface codes with periodic boundary conditions). Such surface codes are very attractive candidates for experimental realization of quantum computing because they can be implemented on a two-dimensional grid of qubits with locality check operators. In these codes, physical qubits are placed on the edges of a two-dimensional lattice of length L, and stabilizers are defined with respect to the code lattice architecture that defines the code space, where k=2, n=2L. 2 is. The stabilizers can be defined in two groups: a vertex operator can be defined at each vertex as the product of X operators on adjacent qubits, and a plaquette operator can be defined at each face as the product of Z operators on the bounding qubits. Thus, a total of 2L 2 There are stabilizers, and each stabilizer group has L 2 There are stabilizers. Assuming that each edge of the lattice has a qubit associated with it, for a given vertex v, the vertex operator is x, as shown in Figure 7. v =Π i∈v X i For a given plaquette p, the plaquette operator Z p =Π i∈v Z i 7 is a schematic diagram of a lattice representation of an exemplary toric code used by a Transformer Neural Network-based decoder, such as decoder 104 (e.g., decoder 104A), in accordance with some embodiments of the present invention.

[0165] Here, experiments are presented that were conducted to evaluate and demonstrate the performance of the Transformer Neural Network Based QECC Decoder 104, specifically the Transformer Neural Network Based QECC Decoder 104A (QECCT). In the experiment, toric codes of various lengths were evaluated while considering two common noise models: independent noise and depolarizing noise. In independent (uncorrelated) noise, X (bit-flip) and Z (phase-flip) errors can occur independently and with equal probability. Therefore, decoding can be performed separately for X or Z stabilizers.

[0166] For depolarizing noise, the equiprobability p / 3 can be assigned to all three Pauli operators

Number

[0167] In an experiment where measurement errors are incorporated, each syndrome measurement is repeated T = n times, and the probability of measurement error is the same as the probability of syndrome error. That is, the distributions of ε and

Number

[0168] As a baseline for performance comparison, the minimum-weight perfect matching (MWPM) decoding algorithm is considered with a complexity of O((n 3 + n 2 ) log(n)). The MWPM decoding algorithm is also known as the Edmonds or Blossom algorithm, which is the most common decoder for topological codes. The MWPM decoder is implemented as known in the prior art and is close to the quadratic mean complexity. In the experiment, the code lengths of the evaluated codes are similar to the code lengths used to test existing end-to-end neural network-based decoders, i.e., code lengths where 2 < L ≤ 10.

[0169] Reference is made to Figures 8A, 8B, 8C, 8D, 8E, and 8F, which are graphical charts comparing the decoding performance of several legacy decoders and a decoder based on a Transformer Neural Network applied to decoding quantum error correcting codes, in accordance with some embodiments of the present invention.

[0170] Both the bit error rate (BER) and the word error rate (LER) are used as metrics to evaluate the performance of the Transformer Neural Network-based QECC decoder 104 (specifically, QECC decoder 104A) (hereinafter referred to as QECCT) and the MWPM algorithm. The LER metric used here is a word-level error metric, meaning that there is an error if at least one quantum bit differs from its true value.

[0171] Graph charts 800 and 802 show the performance of the QECCT compared to the MWPM algorithm for different toric code lengths under independent noise models without noisy measurements. Graph charts 820 and 822 show a similar comparison between the QECCT and MWPM for noisy measurements with T=L and uniformly distributed syndrome errors. Graph charts 830 and 832 show a comparison between the QECCT decoder and the MWPM for noisy and noiseless measurements, respectively, against a depolarized noise model. The graph charts also show the resulting thresholds.

[0172] As can be seen from the graph chart, QECCT significantly outperforms the state-of-the-art MWPM algorithm. First, as is known in the art, MWPM nearly reaches the threshold of state-of-the-art ML algorithms, but QECCT outperforms the MWPM algorithm in independent noise. QECCT also significantly outperforms MWPM in the challenging depolarizing noise setting, with the resulting threshold of 0.178 for QECCT compared to 0.157 for MWPM and 0.189 for ML.

[0173] The very large gap in BER performance may mean that QECCT is better able to detect precise corruption. The threshold is slightly lower for L=10 with depolarized noise, but the BER is much lower, indicating that the regularization parameters may need to be adjusted for larger codes.

[0174] The experiments performed for QECCT go beyond what is typically done for related ML algorithms, in order to determine whether QECCT is applicable to QECC codes other than toric codes.

[0175] Graphical charts 840 and 842 show the performance of the QECCT decoder 104A compared to the MWPM algorithm for different surface code lengths in a depolarized noise model. For the surface code, the same parameters used for the toric code are used. As can be seen, just as QECCT outperforms the MWPM algorithm for the toric code, it also outperforms the MWPM algorithm for the other code, namely, the surface code. The large gap in BER performance in favor of QECCT means that the gap in LER may be even larger with other hyperparameters defined for the loss function (objective).

[0176] To demonstrate generality regarding noise models, graph charts 850 and 852 show the performance of QECCT and MWPM under circuit noise models for different surface code lengths. The transmission channel is simulated using a state-of-the-art STIM simulator of quantum stabilizer circuits, and identical depolarization error probabilities are applied after single- and two-qubit Clifford operations, before stabilizer measurements, and before syndrome measurements. Clearly, QECCT consistently outperforms MWPM even with this type of channel noise.

[0177] Reference is made to Figures 9A, 9B, and 9C, which are graphical charts illustrating the influence of design parameters on the decoding performance of a decoder based on a Transformer Neural Network applied to decode quantum error correcting codes, according to some embodiments of the present invention.

[0178] The graph charts 900 and 902 show the performance of the noise estimator g versus the performance of the QECCT decoder 104A implemented with a neural network architecture with N=6, d=128. ω In particular, the graph charts 900 and 902 show the effect of the noise estimation unit g ω Using the initial scaling (noise) estimate calculated by g ω without using all 1's (i.e., g ω :=1 n ) shows a comparison of the performance of the QECCT using constant mapping and the noise estimation part g ω The impact on performance is shown for two neural network architectures implementing the algorithm, one with one hidden layer (Layers=1) and one with three hidden layers (Layers=3).

[0179] As can be seen, the average test LER of mid-level pooling is 5% lower than that of initial embedding pooling. Clearly, the initial noise estimation part g ω is important for performance. However, it is less affected by the network architecture. Given more resources, additional architectures can be explored.

[0180] The graph chart 902 may show the effects of different pooling (averaging) scenarios. Pooling may be performed after the initial embedding in the input layer 302 (denoted in the graph as "Pooled Initial Emb."), ​​in the middle of the neural network in the decoding layer 304 (denoted in the graph as "Pooled Mid Emb."), ​​or in the final embedding in the output layer 306 (denoted in the graph as "Pooled Final Emb."). Clearly, pooling in the middle layers can significantly improve decoding performance, while pooling the final embedding (as with majority voting) is ineffective.

[0181] Graphical charts 910 and 912 show the impact of various loss functions (objectives) used in Equation 11. In particular, graphical chart 910 shows the effect of the loss function L on the overall loss function. BER , L LER , and L g Each of these has its own weight λ BER , λ LER , and λ g , and a graphical chart 920 shows the effect of regularization on the training dynamics for a QECCT decoder 104A employing a neural network architecture with N=6, d=128.

[0182] First, the decoder 104A is LER It can be observed that training only on the loss function (objective) can rapidly converge to a bad local minimum. This is illustrated in the graphical chart 912 by displaying the mean gradient norm across the network layers. BER When training with only L LER This gives much worse results than combining it with a loss function. BER A QECCT decoder trained with the loss function L can yield 27 times higher LER compared to the same QECCT decoder trained with the combined loss function. gBy optimizing with , we obtain a 46% improvement at high SNR compared to no regularization.

[0183] The graph chart 920 shows the impact of using a mask, i.e., a mask derived from the QECC code, on the performance of the QECCT decoder 104A implemented with an N=6, d=128 neural network architecture, particularly with respect to the masking capacity of the QECCT decoder 104A and the straight-through estimator (STE) as a bin function from Equation 10. Thus, although not as important as with classical codes known in the prior art, the mask still has a significant impact on performance.

[0184] The graph chart 922 shows the effect of model size, i.e., the architecture of the Transformer Neural Network that implements the QECCT decoder 104A, on its decoding performance. It can be seen that increasing the capacity of the Transformer Neural Network decoder 104A, i.e., increasing the number of its decoding layers, can result in better representation and potentially improved decoding performance.

[0185] The description of various embodiments of the present invention has been presented for purposes of explanation, but is not intended to be limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the embodiments described herein. The terms used herein have been selected to best explain the principles of the embodiments, practical applications or technical improvements to technology found in the market, or to enable those skilled in the art to understand the embodiments disclosed herein.

[0186] It is expected that many related systems, methods, and computer programs will be developed during the life of the patent that matures from this application, and the scope of the terms error correcting code, quantum error correcting code, and error correcting code decoding model is intended a priori to include all such new technologies.

[0187] As used herein, "about" refers to ±10%.

[0188] The words "comprises," "comprising," "includes," "including," "having," and conjugations thereof, mean "including but not limited to." This term encompasses the terms "consisting of" and "consisting essentially of."

[0189] The term "consisting essentially of" means that a composition or method may include additional ingredients and / or steps, provided that the additional ingredients and / or steps do not materially alter the basic and novel characteristics of the claimed composition or method.

[0190] As used herein, the singular forms "a," "an," and "the" include plural references unless the context clearly dictates otherwise. For example, "a compound" or "at least one compound" includes a plurality of compounds, and may also include mixtures thereof.

[0191] "Exemplary" is used herein to mean "serving as an example, instance, or illustration." Any embodiment described as "exemplary" is not necessarily to be construed as preferred or advantageous over other embodiments and / or to exclude the incorporation of features from other embodiments.

[0192] "Optionally" is used herein to mean "provided in some embodiments and not provided in other embodiments." Any particular embodiment of the present invention may include multiple "optional" features unless those "optional" features contradict each other.

[0193] Throughout this application, various embodiments of the invention may be presented in a range format. It should be understood that the description in range format is merely for convenience and brevity and is not an inflexible limitation on the scope of the invention. Thus, the description of a range should be considered to specifically disclose all of the possible subranges and individual numerical values ​​within that range. For example, description of a range such as 1 to 6 specifically discloses subranges such as 1 to 3, 1 to 4, 1 to 5, 2 to 4, 2 to 6, 3 to 6, etc., as well as individual numerical values ​​within that range, e.g., 1, 2, 3, 4, 5, and 6. This applies regardless of the magnitude of the range.

[0194] Whenever a range of numerical values ​​is given herein, it is intended to include any recited number (fractional or integer) within the range given. The phrases "ranging between" a first designated number and a second designated number and "ranging from" a first designated number to a second designated number are used interchangeably herein and are intended to include the first designated number and the second designated number, and all fractional and integer numbers therebetween.

[0195] It will be understood that certain features of the invention, which are, for clarity, described in the context of separate embodiments, may also be provided in any combination of these features in a single embodiment. Conversely, multiple features of the invention, which are, for brevity, described in the context of a single embodiment, may also be provided separately or in any suitable subcombination or with respect to other described embodiments as appropriate. A given feature described in the context of various embodiments should not be construed as essential to that embodiment, unless the particular embodiment is inoperable without that element.

[0196] While the present invention has been described in conjunction with specific embodiments thereof, many alternatives, modifications, and variations will be apparent to those skilled in the art. Accordingly, it is intended to embrace all such alternatives, modifications, and variations that fall within the spirit and broad scope of the appended claims.

[0197] It is the intention of the applicants that all publications, patents, and patent applications mentioned in this specification be incorporated herein by reference in their entirety to the same extent as if each individual publication, patent, and patent application was specifically and individually indicated to be incorporated herein by reference. In addition, citation or identification of any reference in this application should not be construed as an admission that such reference is available as prior art to the present invention. Nor should it necessarily be construed as limiting, to the extent that section headings are used. In addition, the priority document of this application, if any, is incorporated herein by reference in its entirety.

Claims

1. 1. A transformer neural network based decoder for decoding quantum error correcting codes, comprising: an input layer adapted to receive an initial noise estimate calculated by a noise estimator of noise injected into syndrome bits of at least one codeword encoded using a quantum error correcting code and transmitted over an interference-subjected transmission channel, and to create an embedding of the syndrome bits; a plurality of decoding layers adapted to calculate an estimated logical operator matrix for the at least one codeword, each of the plurality of decoding layers including a self-attention layer including at least one head constructed according to a mask indicating a relationship between the embeddings, the relationship between the embeddings being derived from a parity check matrix of an error correcting code such that the mask is adapted to unmask pairs of connected parity bits and to mask pairs of disconnected parity bits; an output layer adapted to generate a vector representing a predicted soft error of the logical operator matrix of the at least one codeword; The plurality of decoding layers are trained using a joint loss function that aims to minimize a logarithmic error rate (LER), a bit error rate (BER), and an error rate of the noise estimator. Decoder based on Transformer Neural Networks.

2. the mask is created based on an extended bipartite graph representation of the parity check matrix of the error correcting code, the bipartite graph representation including a plurality of nodes connected via a plurality of edges, each pair of connected bits including a bit that shares at least one of the plurality of nodes, and each pair of unconnected bits including a bit that does not share any of the plurality of nodes; 10. A transformer neural network based decoder according to claim 1.

3. the bipartite graph is a Tanner graph, 3. A transformer neural network based decoder according to claim 2.

4. Each of the plurality of decoding layers further includes a feedforward layer interleaved with a normalization layer from the self-attention layer.

10. A transformer neural network based decoder according to claim 1.

5. the embedding produced by the input layer has a higher dimensionality than the dimensionality of the received initial noise estimate; 10. A transformer neural network based decoder according to claim 1.

6. the output layer is configured to reduce the dimension of a soft error vector obtained by concatenating multiple soft error vectors calculated by the multiple decoding layers based on the embedding.

10. A transformer neural network based decoder according to claim 1.

7. each term of the joint loss function is adjusted according to a weighting assigned to each of the LER, the BER, and the error rate of the noise estimator; 10. A transformer neural network based decoder according to claim 1.

8. the parity check matrix includes a bit-flip parity check matrix calculated to correct bit flips of quantum bits, and a phase-flip parity check matrix calculated to correct phase flips of quantum bits; 10. A transformer neural network based decoder according to claim 1.

9. the noise estimator is implemented using at least one shallow neural network parameterized during training using a plurality of training samples including a plurality of noise-injected sets of syndrome bits; 10. A transformer neural network based decoder according to claim 1.

10. The quantum error correcting code is one of the group consisting of a stabilizer code, a surface code, and a topological code.

10. A transformer neural network based decoder according to claim 1.

11. 1. A method for training a Transformer Neural Network based decoder for decoding quantum error correcting codes, comprising: Using at least one processor, obtaining a plurality of training samples, the training samples comprising a plurality of initial noise estimates calculated by at least one noise estimator, of noise injected into syndrome bits of at least one codeword encoded with at least one quantum error correcting code and transmitted over at least one transmission channel subject to interference; A decoder based on a transformer neural network calculating an estimated Boolean operator matrix for the at least one codeword by minimizing a joint loss function that aims to minimize a Logical Error Rate (LER), a Bit Error Rate (BER), and an error rate of the noise estimator; and by generating a vector representing predicted soft errors in the logical operator matrix of the codeword; using the plurality of training samples to train the Transformer neural network-based decoder to decode codewords encoded with the at least one quantum error correcting code; outputting a decoder based on the trained Transformer neural network for decoding at least one codeword encoded with at least one quantum error correcting code; A method comprising:

12. the loss function for the LER is defined based on a binary cross-entropy loss for the predicted soft errors calculated by the trained Transformer neural network-based decoder; The method of claim 11.

13. The method further includes redefining the LER loss function as a differentiable loss function using a differentiable equivalent mapping of an XOR operation based on a binary mod 2, thereby enabling minimization of the LER differentiable loss function. The method of claim 12.

14. further comprising defining a differentiable loss function of the LER based on a binarization of the predicted soft errors. The method of claim 13.

15. further comprising regularizing the differentiable loss function of the LER. The method of claim 12.

16. the at least one noise estimator includes at least one shallow neural network trained to calculate the initial noise estimate by minimizing a binary cross-entropy loss for syndrome bits using a plurality of training samples including syndrome bits of the at least one codeword encoded using the at least one quantum error correcting code. The method of claim 11.

17. the at least one encoded codeword used to create the training sample includes a zero codeword. The method of claim 11.

18. 1. A method of using a decoder based on a transformer neural network for decoding a quantum error correcting code, comprising: receiving an initial noise estimate, calculated by a noise estimator, of noise injected into syndrome bits of at least one codeword encoded with a quantum error correcting code and transmitted over an interference-affected transmission channel; applying a trained neural network based decoder to calculate an estimated logical operator matrix for the at least one codeword and to generate a vector representing predicted soft errors in the logical operator matrix; and outputting a vector representing the predicted soft errors; The trained neural network based decoder comprises: an input layer adapted to receive the initial noise estimate and to create an embedding of the syndrome bits; a plurality of decoding layers adapted to calculate an estimated Boolean operator matrix for the at least one codeword according to a mask indicating a relationship between the embeddings; an output layer adapted to generate a vector representing the predicted soft error; method.

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