Distributed machine learning via quantum computing and quantum communication

Abstract

One embodiment includes a method comprising receiving a first input in a distributed quantum computing system (DQCS) having a plurality of quantum nodes and a quantum communication network. A first quantum computation is performed on a first quantum node among the plurality of nodes based on the first input and a model. The output of the first quantum computation is communicated to a second quantum node among the plurality of quantum nodes via the quantum communication network. The results of the distributed computation are based on the output of the first quantum computation and a quantum tomography algorithm performed on the second quantum node.

Description

TECHNICAL FIELD

[0001] Cross-Reference to Related Applications This application claims priority to U.S. Provisional Patent Application No. 63 / 583,108, filed on September 15, 2023, entitled "DISTRIBUTED MACHINE LEARNING VIA QUANTUM COMPUTATION AND QUANTUM COMMUNICATION", the entire content of which is incorporated herein by reference in its entirety.

[0002] The present disclosure relates generally to quantum computing systems, and more specifically to distributed machine learning via quantum computation and quantum communication.

BACKGROUND ART

[0003] Quantum computing is a computing method that uses quantum effects such as superposition of ground states and entanglement to perform specific calculations more efficiently than classical digital computers. In contrast to digital computers that store and manipulate information in the form of bits, for example "1" or "0", quantum computing systems can use quantum bits ("qubits") to manipulate information. A qubit may refer to a quantum device that provides a superposition of data in multiple states, for example both the "0" and "1" states, and / or the superposition itself of data in multiple states. According to conventional terminology, the superposition of the "0" and "1" states in a quantum system can be expressed, for example, as a|0⟩+b|1⟩. The "0" and "1" states of a digital computer correspond to the |0⟩ and |1⟩ ground states of a qubit, respectively.

SUMMARY OF THE INVENTION

MEANS FOR SOLVING THE PROBLEMS

[0004] Aspects and advantages of the embodiments of the present disclosure are set forth in part in the description that follows, or can be learned from the description, or can be learned through practice of the embodiments.

[0005] One embodiment includes a method comprising receiving a first input at a distributed quantum computing system (DQCS) having a plurality of quantum nodes and a quantum communication network. A first quantum computation is executed at a first quantum node among the plurality of nodes based on the first input and a model. An output of the first quantum computation is communicated to a second quantum node among the plurality of quantum nodes via the quantum communication network. A result of the distributed computation is based on the output of the first quantum computation and a quantum tomography algorithm performed at the second quantum node.

[0006] Other aspects of the present disclosure are directed to various systems, methods, apparatuses, non-transitory computer-readable media, computer-readable instructions, and computing devices.

[0007] These and other features, aspects, and advantages of various embodiments of the present disclosure will be better understood with reference to the following description and appended claims. The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate exemplary embodiments of the present disclosure and together with the description explain the related principles.

[0008] The detailed discussion of the embodiments directed to those skilled in the art is set forth herein with reference to the following accompanying drawings. [BRIEF DESCRIPTION OF THE DRAWINGS]

[0009] [Figure 1] Illustrates an exemplary quantum computing system in accordance with exemplary embodiments of the present disclosure. [Figure 2A] Shows a non-limiting example of a distributed computing environment in which various embodiments may be applied. [Figure 2B] Illustrates a schematic diagram of bandwidth requirements for both classical communication protocols and quantum communication protocols when calculating gradients for various embodiments. [Figure 3] Illustrates a schematic diagram of a distributed quantum circuit in accordance with various embodiments. [Figure 4A]We present non-restrictive examples of the first algorithm, a quantum tomography algorithm, in various embodiments. [Figure 4B] This document presents a non-restrictive second algorithm for distributed stochastic coordinate descent, encompassing various embodiments. [Figure 5] Flowcharts illustrate exemplary methods for performing distributed computation via models and distributed quantum computing systems, according to various embodiments. [Modes for carrying out the invention]

[0010] Exemplary aspects of this disclosure relate to the implementation of machine learning (e.g., both model training and deployment of trained models) via distributed computing. More specifically, embodiments implement distributed quantum computing in the context of training and deploying machine learning models. In the overall description, various machine learning trained models may be implemented via one or more distributed neural network architectures. Embodiments are not limited in this way, but other distributed architectures may be used to implement machine learning models of various embodiments. A quantum communication protocol is implemented to establish quantum communication channels between nodes within a distributed quantum computing asset. The quantum communication protocol leverages quantum shadow tomography so that the number of qubits that must be communicated between quantum computing nodes is exponentially smaller than the number of neural network parameters and the size of the input (training or runtime).

[0011] As the scale of datasets and parameterized models used to perform computations on data continues to expand, distributing the workload across hardware becomes useful in enabling progress. This necessitates considering not only computational complexity but also the fact that communication can become a bottleneck in large-scale training and inference. While often discussed in the context of computational superiority, quantum technology also offers the exponential reduction of the amount of data that needs to be transferred for some tasks, and these communication superiorities could dramatically impact future distributed learning models. While neural network architectures are constantly evolving, one constant feature when using them is the crucial role of gradient-based optimization in fitting these models to data. This typically requires communication of data, features, and / or gradients between nodes performing parameter updates. Using models for inference on user-provided data also requires communication of either the model or the input, which can itself be quite large. Large pre-trained models are often fine-tuned to solve downstream tasks, which involves adding additional layers to the end of the pre-trained model and performing additional training. This requires communication of either the model itself or the input and / or features. For a fairly common class of learning models, it is shown below that if quantum bits (also called qubits) are used for communication instead of classical bits, an exponential reduction in the communication required to solve such problems can be achieved. This protocol further guarantees the privacy of both user data and model parameters through the natural features of quantum mechanics, without requiring additional encryption or privacy protocols. The advantages of these types of communication help define the future role and scope of interaction between quantum and classical communication in distributed machine learning.

[0012] More specifically, the size of the neural network and the dataset on which the neural network is trained are continuously scaled up because scaling tends to improve performance. Beyond a certain point, training may be performed in a distributed manner across multiple machines / groups of machines. As mentioned above, this may require communication of the data or internal features learned by the model between workers in order to train the model or, if trained, to use it for inference. By using distributed quantum computing and quantum communication methods, embodiments reduce the communication costs when training a distributed parameterized model using gradient descent and when using the trained model for inference. Thus, embodiments facilitate training on datasets that are too large to store centrally.

[0013] The embodiments can be used in any configuration in which classical computers are used to store distributed neural network models and training data. In the embodiments, classical computers (e.g., distributed among sets of nodes) are replaced by quantum computers (e.g., distributed among sets of nodes). Furthermore, in the embodiments, instead of a classical network connecting the nodes, quantum communication channels connect the quantum computers of the set of nodes in a communicative manner. The quantum channels are configured to transmit coherent quantum states between nodes. In these configurations, the embodiments reduce the task of computing the output of the network (e.g., given several inputs) and the task of computing the gradient of the network parameters to a quantum shadow tomography method. The embodiments establish a lower bound on the number of copies of the quantum state required for shadow tomography and transmit that number of copies of the coherent quantum state between nodes. It is shown that the number of qubits that must be communicated between quantum computers is exponentially smaller than the number of parameters in the network and the size of the inputs.

[0014] Embodiments may be used to train large-scale distributed models using gradient descent. Embodiments may also be used during inference of large-scale models, for example, when a large-scale trained model is presented with user data. Embodiments may also be used to fine-tune a trained model by adding additional layers. All these scenarios exhibit compositional structures that benefit from quantum communication, as presented herein.

[0015] In addition to reducing communication complexity via quantum communication channels and quantum shadow tomography, the embodiments also offer the additional advantage of data privacy. This advantage of data privacy arises from the disruptive nature of quantum measurement. This means that the communication necessary to solve these problems is inherently more private than conventional communication, because even if all communication is intercepted by a third party, only a logarithmic amount of information can be extracted with respect to the input data. This is useful for ensuring the privacy of both user data and the parameters of trained models.

[0016] From a determined lower bound on the communication complexity of the related problem, it can be shown that, in at least some settings, the amount of classical communication must be substantially proportional to the size of the input in order to solve this problem. The embodiment exponentially reduces communication complexity compared to standard methods of training and inference using neural networks.

[0017] One embodiment includes a method comprising receiving a first input in a distributed quantum computing system (DQCS) having a plurality of quantum nodes and a quantum communication network. A first quantum computation is performed on a first quantum node among the plurality of nodes based on the first input and a model. The output of the first quantum computation is communicated to a second quantum node among the plurality of quantum nodes via the quantum communication network. The results of the distributed computation are based on the output of the first quantum computation and a quantum tomography algorithm performed on the second quantum node.

[0018] Aspects of this disclosure offer several technical effects and advantages. For example, embodiments configure the training and deployment of large-scale machine learning models distributed across multiple quantum computing nodes. The quantum nodes communicate with each other using quantum shadow tomography. Shadow tomography significantly reduces communication complexity because the amount of qubits that need to be transferred between nodes decreases exponentially with respect to the size of the model and training datasets. This exponential prediction in communication complexity provides distribution for large models (e.g., model training, model fine-tuning, and inference time) compared to classical methods. Embodiments also offer the data privacy advantages inherent in quantum communication channels.

[0019] Figure 1 shows an exemplary quantum computing system 100. System 100 is an example of a system of one or more classical computers and / or quantum computing devices in one or more locations, and can implement the systems, components, and techniques described below. Those skilled in the art will understand that other quantum computing devices or systems can be used with the disclosures provided herein without departing from the scope of this disclosure.

[0020] System 100 includes quantum hardware 102 and one or more classical processors 104 that communicate data with the quantum hardware 102. The classical processors 104 can be configured to execute computer-readable instructions stored in one or more memory devices to perform any of the operations described herein. The quantum hardware 102 includes components for performing quantum computation. For example, the quantum hardware 102 includes a quantum system 110, a control device 112, and a readout device 114 (e.g., a readout resonator). The quantum system 110 may include one or more multi-level quantum subsystems, such as registers for qubits (e.g., qubit 120). In some embodiments, the multi-level quantum subsystems may include superconducting qubits such as flux qubits, charge qubits, transmon qubits, gmon qubits, and spin-based qubits.

[0021] The type of multi-level quantum subsystem utilized by system 100 can vary. For example, it may be advantageous in some cases to include one or more superconducting qubits, e.g., transmon, fluxmon, gmon, xmon, or one or more readout devices 114 attached to other qubits. In other cases, ion traps, photonic devices, or superconducting cavities (e.g., those capable of preparing states without requiring qubits) may be used. Further examples of realizations of multi-level quantum subsystems include fluxmon qubits, silicon quantum dots, or phosphorus impurity qubits.

[0022] A quantum circuit can be constructed and applied to the registers of qubits contained in a quantum system 110 via a plurality of control lines coupled to one or more control devices 112. An exemplary control device 112 acting on the qubit registers may be used to implement a quantum circuit having a quantum gate, or a plurality of quantum gates, such as a Pauli gate, Hadamard gate, controlled NOT (CNOT) gate, controlled phase gate, T gate, multi-qubit quantum gate, coupler quantum gate, etc. One or more control devices 112 may be configured to act on the quantum system 110 through one or more respective control parameters (e.g., one or more physical control parameters). For example, in some embodiments, the multi-level quantum subsystem may be a superconducting qubit, and the control device 112 may be configured to provide control pulses to the control lines to generate a magnetic field that adjusts the frequency of the qubit.

[0023] The quantum hardware 102 may further include a readout device 114 (e.g., a readout resonator). Measurement results 108 obtained via the measurement device may be provided to a classical processor 104 for processing and analysis. In some embodiments, the quantum hardware 102 may include quantum circuits, and the control device(s) 112 and readout device(s) 114 may implement one or more quantum logic gates operating on the quantum system 100 using physical control parameters (e.g., microwave pulses) transmitted via wires included in the quantum hardware 102. Further examples of control devices include any waveform generator in which a DAC (digital-to-analog converter) generates a signal.

[0024] The readout device(s) 114 may be configured to perform quantum measurements on the quantum system 110 and transmit the measurement results 108 to the classical processor 104. Furthermore, the quantum hardware 102 may be configured to receive data from the classical processor 104 specifying physical control qubit parameter values ​​106. The quantum hardware 102 may use the received physical control qubit parameter values ​​106 to update the actions of the control device(s) 112 and readout device(s) 114 on the quantum system 110. For example, the quantum hardware 102 may receive data specifying a new value representing the voltage intensity of one or more DACs contained within the control device 112, and accordingly update the action of the DACs on the quantum system 110. The classical processor 104 may be configured to initialize the quantum system 110 to an initial quantum state by, for example, transmitting data specifying an initial set of parameters 106 to the quantum hardware 102.

[0025] In some embodiments, the readout device 114 can measure the state of an element of a quantum system, such as a qubit, by utilizing the difference in impedance between the |0> and |1> states of the element. For example, due to the nonlinearity of the qubit, the resonant frequency of the readout resonator may be different when the qubit is in the |0> or |1> state. Therefore, the microwave pulses reflected from the readout device 114 have amplitude and phase shifts corresponding to the qubit state. In some embodiments, a parcel filter may be used in conjunction with the readout device 114 to obstruct microwave propagation at the qubit frequency.

[0026] In some embodiments, the quantum system 110 may include a plurality of qubits 120 arranged, for example, in a two-dimensional grid 122. For clarity, the two-dimensional grid 122 depicted in Figure 1 contains 4x4 qubits, but in some embodiments, the system 110 may contain fewer or more qubits. In some embodiments, the plurality of qubits 120 may interact via a plurality of qubit couplers, for example, a qubit coupler 124. The qubit coupler can define nearest-neighbor interactions between the plurality of qubits 120. In some embodiments, the strength of the plurality of qubit couplers is a tunable parameter. In some cases, the plurality of qubit couplers included in the quantum computing system 100 may be couplers with a fixed coupling strength.

[0027] In some embodiments, the qubits 120 may include data qubits such as qubit 126 and measurement qubits such as qubit 128. Data qubits are qubits involved in calculations performed by the system 100. Measurement qubits are qubits that can be used to determine the results of calculations performed by the data qubits. That is, during calculations, the unknown state of the data qubits is communicated to the measurement qubits using appropriate physical actions and measured by appropriate measurement actions performed on the measurement qubits.

[0028] In some embodiments, each of the multiple qubits 120 may operate using its own operating frequency, such as an idling frequency and / or interaction frequency and / or read frequency and / or reset frequency. The operating frequency may vary from qubit to qubit. For example, each qubit may idle at a different operating frequency. The operating frequency of qubit 120 may be selected before the calculation is performed.

[0029] Figure 1 shows an example of a quantum computing system that may be used to carry out the methods and operations according to exemplary embodiments of this disclosure. Other quantum computing systems may be used without departing from the scope of this disclosure.

[0030] Training and inference with large-scale machine learning models that far exceed the memory capacity of individual devices may require the design of distributed architectures, necessitating the address of communication constraints. The embodiment includes a framework for distributed computation via a quantum network where data is encoded into special quantum states. The embodiment provides machine learning (ML) quantum models within this framework. Gradient descent is used to train these models. Within the quantum framework of the embodiment, both inference and gradient descent of these models can be performed with exponentially less communication compared to classical analogues and with relatively moderate overhead compared to standard (classical) gradient-based methods. Several graph neural networks are particularly well-suited for implementation within this framework. That is, the embodiment provides an exponential quantum advantage over a class of general machine learning problems that hold regardless of data encoding costs. Furthermore, this class of models can encode highly nonlinear features of the input, and their expressive power increases exponentially with the model depth. Due to the inherent properties of quantum states, the embodiment provides a natural privacy advantage in communicated quantum states, limiting the amount of information that can be extracted from the data and model parameters. The embodiment provides a foundation for distributed machine learning via quantum networks.

[0031] As the scale of datasets and parameterized models used to perform computations on data continues to expand, distributing workloads across multiple devices can be useful in enabling progress. Choosing an architecture for large-scale training and inference must address not only the maximum utilization of computational and memory resources, but also the fact that communication can become a bottleneck. This is particularly important as models become so large that they cannot rely on high-bandwidth interconnects within a single data center, and instead are trained across multiple data centers. When using modern optical interconnects, classical computers exchange bits represented by light. However, this does not fully utilize the potential of the physical substrate. Given the appropriate computational power and algorithms, the quantum properties of light can be leveraged as a powerful communication resource. Embodiments provide a broad class of parameterized models in which quantum bits (qubits) are communicated instead of classical bits, achieving an exponential reduction in the communication required to perform inference and gradient-based training. This protocol further ensures enhanced privacy for both user data and model parameters through the natural properties of quantum mechanics, without requiring additional encryption or privacy protocols. The embodiments provide a general exponential quantum advantage over problems that naturally arise in training and deploying large-scale machine learning models. The advantages of these types of communication help define the future role and scope of interaction between quantum and classical communication in distributed machine learning.

[0032] Quantum computers can offer dramatic speedups across many computational tasks. One notable example of these tasks is their ability to revolutionize our understanding of nature by enabling the simulation of quantum systems, given the natural similarity between quantum computers and the real world. However, much of the data we want to compute in practice appears to originate from the classical world in which it emerges, rather than directly exhibiting quantum properties. While there are several examples of exponential quantum speedups over classical problems, the most prominent being factorization and related hidden subgroup problems, these tend to be isolated and sometimes difficult to relate to practical applications that learn from data. Furthermore, even if significant speedups are known for certain ubiquitous problems in machine learning, such as matrix inverse calculation and principal component analysis, the advantage is often lost when the cost of loading classical data into a quantum computer or reading the results into classical memory is included. This is because the complexity of loading dense classical data into the amplitudes of quantum states (which is usually the encoding required to gain the advantage of exponential runtime) and the complexity of reading the amplitudes from quantum states into classical memory are both polynomials of the number of amplitudes. In applications where the above risks are avoided by efficient data access models, the complexity of quantum algorithms tends to depend on the condition number of the matrix, and since this condition number scales with system size, the quantum advantage may be reduced or even completely lost. It is worth noting that much of the discussion on the impact of quantum technology on machine learning focuses on computational advantages. However, quantum resources are not only useful for reducing computational complexity, but can also provide an advantage in communication complexity, enabling exponential reductions in communication for some problems. Embodiments leverage the advantages of quantum communication across a broad class of machine learning models. This advantage is established without requiring any sparsity assumptions or complex data access models.

[0033] Figure 2A shows a non-limiting example of a distributed computing environment 200 to which various embodiments may be applied. The distributed computing environment 200 includes a set of computing resources distributed across space. Without loss of generality, the distributed computing environment 200 includes a first resource set 210, a second resource set 212, a third resource set 214, and a fourth resource set 216. Resource sets are sometimes referred to as resource nodes (or simply nodes). Various resource sets may include classical and / or quantum computing resources. Thus, a resource set may be a set of computing resources. When a resource set (or node) includes quantum computing hardware, the resource set (or node) is sometimes referred to as a quantum node (or quantum resource set). Therefore, the first resource set 210 may be called the first quantum node, the second resource set 212 may be called the second quantum node, the third resource set 214 may be called the third quantum node, and the fourth resource set 216 may be called the fourth quantum node. The distributed computing environment 200 may be implemented by a distributed classical computing system (DCCS), a distributed quantum computing system (DQCS), or a combination thereof.

[0034] As shown via coordinate system 208, each resource set may be spatially isolated from each of the other resource sets. Furthermore, resource sets may be communicably coupled by classical and / or quantum communication networks or channels, as shown by lines connecting various operations of the distributed computing environment 200 (across both time and space). That is, lines connecting various operations may represent communication (e.g., classical and / or quantum communication) between each set of node sets (or nodes). The distributed computing environment 200 includes an input set 202, a compute set 204, and an output set 206. As shown in Figure 2A, the input set 202, compute set 204, and output set 206 are distributed in time and space. Also, communication between the input set 202, compute set 204, and output set 206, as well as between resource sets, can be inherently classical, inherently quantum, or a combination thereof. The circular nodes of the compute set may be functions (e.g., parameterized functions) and / or hardware resources allocated to compute functions. The rectangular nodes in input set 202 and output set represent data (e.g., classical and / or quantum data). If the computation in computation set 204 involves processing quantum information, then that computation may be a quantum computation.

[0035] The following description focuses on constructive distributed learning (e.g., pipelining). The embodiments are not limited to constructive distributed learning, and other embodiments may be generalizable to other forms of distributed learning. For example, several strategies exist for distributing machine learning workloads, which are affected by different application requirements and hardware constraints. Figure 2A shows a non-restrictive example of partitioning a computation graph in a constructive manner. Distributed parameterized quantum circuits can be described in the context of a distributed computing environment. Such distributed quantum circuits, when distributed in this manner, can be used to perform inference on data and can be trained using gradient descent. Embodiments can also be used to optimize models that use certain forms of data parallelism. In principle, such circuits can be implemented in a quantum computer capable of communicating quantum states.

[0036] Even with simple distributed quantum circuits, there is an exponential quantum advantage in communication over the problem of estimating parameter losses and loss gradients. This further implies a privacy advantage. Embodiments use a class of models that can efficiently approximate specific graph neural networks. The models in embodiments not only maintain an exponential advantage in communication but also achieve better (or at least equivalent) performance than standard classical models on common node and graph classification benchmarks. Some distributed quantum circuits in embodiments have an exponential advantage in communication not only over a single round of gradient estimation but throughout the entire training process. This includes circuits for fine-tuning using pre-trained features. Some embodiments interleave multiple unitaries that encode nonlinear features of the data. The ability to interleave multiple unitaries allows for an exponential increase in expressiveness with depth and, in some settings, enables universal function approximation. Thus, the models in embodiments have high expressiveness, in contrast to the common notion of linear constraints in quantum neural networks.

[0037] Some embodiments use a pipelining method to distribute machine learning workloads, where different layers of a deep model are assigned to separate hardware resources. Training and inference then require feature communication between nodes (or resource sets). Pipelining allows for the adaptive combination of subsets of a large model to solve many downstream tasks, thus enabling flexible modification of the model architecture depending on the task. Furthermore, pipelining facilitates better use of heterogeneous computing resources by enabling sparse activation of the subset of the model needed to solve a task and eliminating the need to store identical copies of the large model. Through pipelining, large (and distributed) models can be fine-tuned to solve multiple tasks. In contrast to pipelining, data parallelism involves storing multiple copies of a model on different nodes, training each on a subset of data, and exchanging information to synchronize parameter updates. In practice, different parallelization strategies are combined to leverage the trade-off between latency and throughput depending on the task.

[0038] Communication complexity is the study of distributed computing problems using cost models that focus on the communication required between players, rather than the complexity of time or computation. The main subject of study in this area is a tree in which nodes enumerate all possible communication histories, and leaves are induced by communication protocols corresponding to the outputs of protocols. The product structure induced in the leaves of this tree as a function of inputs allows for a constraint on the depth of the tree from below, thereby providing an unconditional lower bound for communication complexity. The ability to replace classical communication bits with qubits has been studied. In certain problems such as hidden matching and classification deformations using deep linear models, the exponential superiority of quantum communication is maintained, but in canonical problems such as separability, only polynomial superiority is possible.

[0039] Exponential superiority has been demonstrated for problems that sample from a distribution defined by the solutions of linear regression problems. Many models exist for both quantum and classical communication, and embodiments apply to many of these models. For example, several unrestricted embodiments apply to randomized classical communication complexity. In some studies of these embodiments, nodes are allowed to swap random bits (of information) independently of the problem's input, and are also allowed to output incorrect answers with a certain probability (in the case of binary output problems, a range far from 0.5). It should also be noted that the superiority of the type of communication shown here can naturally relate to the spatial superiority of streaming algorithms, which may also be of interest in settings that do not involve distributed training.

[0040] Embodiments may utilize networked quantum computers. For some quantum network architectures of embodiments, the presence of long-term fault-tolerant quantum memory as quantum repeaters can be an effective component for increasing low-rate shared entanglement in a fully functional quantum network. Therefore, when discussing applications of fault-tolerant quantum computers, it is well motivated to consider potential communication advantages along with computational advantages.

[0041] Figure 2B shows schematic diagrams of bandwidth requirements for both classical and quantum communication protocols when calculating gradients for various embodiments. More specifically, Figure 2B shows a schematic diagram of classical communication for calculating gradient 220 during a training epoch. Figure 2B also shows a schematic diagram of quantum communication for calculating gradient 230 during a training epoch. In Figure 2B, g i This shows the gradient estimators implemented in each resource set, μ i This shows the forward features computed in each resource set, v iThis shows the inverse features computed in each resource set. Thus, the forward and inverse paths in each classical communication for computing the gradient 220, and the forward and inverse paths in each quantum communication for computing the gradient 230, are represented by the temporal sequence of forward and inverse features. In the classical communication for computing the gradient 220, L represents the classical loss function, while in the quantum communication for computing the gradient 230, P0 represents the quantum operator that gives a similar loss function when the expectation value for the quantum model is quantum.

[0042] The size of the arrows between nodes in the calculation of forward and reverse features indicates the relative "size difference" of the amount of information that needs to be communicated between the nodes. For example, a larger arrow indicating the bandwidth required for classical information 222 in classical communication to calculate gradient 220 is much larger than a smaller arrow indicating the bandwidth required for quantum information 232. In some embodiments, the bandwidth requirement for quantum information is exponentially smaller than the bandwidth requirement for classical information.

[0043] Formal theory of distributed learning using quantum resources Figure 3 shows schematic diagrams of distributed quantum circuit 300 in various embodiments. For simplicity, distributed quantum circuit 300 assumes only two nodes in the distributed computing environment. However, in general, distributed quantum circuit 300 can include many more nodes. Also, in Figure 3 (and the whole), canonical communication nomenclature is used for the two communicating parties, and the two nodes are called Alice and Bob (e.g., two spatially distributed parties exchanging (quantum) information). Thus, throughout, resource sets (or nodes) can be called “players” or “communicating parties”. Coordinate system 308 is provided to distributed quantum circuit 300. Note that the axes of coordinate system 308 are swapped with those of coordinate system 208 in Figures 2A-B. Overall, L is an unbounded positive integer indicating the number of rounds of communication between nodes (or players). In a non-restrictive example of distributed quantum circuit 300, L=2. Embodiments are not so limited, and embodiments where L≫2 are also contemplated.

[0044] In FIG. 3, communication problems with two nodes (e.g., players or parties), Alice and Bob, are considered. More specifically, Alice and Bob have a normalized data input x∈S of length N N-1 and A l , B l defined by an N×N unitary matrix that defines a model given by ∈U(N), seek to compute a coupling function determined through L rounds of communication. The data matrix and the model matrix are distributed between Alice and Bob as follows. Alice: x, {A l} (Equation 1a) Bob: {B l}, (Equation 1b)

[0045] Without loss of generality, A l , B l are each parameterized by P l parameters, and it is assumed that P≡max l P l . For simplicity, these are denoted for A l as [Mathematical formula] , in some embodiments, [Mathematical formula] is also assumed to be unitary, and the same applies to B l . While seemingly strict, this is, for example, satisfied in many of the most common quantum neural network architectures when [Mathematical formula] is unitary and Hermitian (e.g., Pauli matrices). Such models, or parameterized quantum circuits, are naturally suitable for implementation on quantum devices. Embodiments consider a general class of models of the following formula. [Mathematical formula]

[0046] In some embodiments, the quantity (or the gradient of the next quantity) <φ|P0|φ> ≡ L (Equation 3)

[0047] We estimate the following. In the equation, P0 is the Pauli matrix that acts on the first qubit after L rounds of communication, as shown in Figure 3. ∇ A(B) In <φ|Z0|φ>, among the gradient vectors, {A l}({B l This represents the element corresponding to}).

[0048] At a high level, the state in Equation 3 is a model of multi-program multi-data (MPMD) or pipelining computation where different stages of computation with a constitutive structure are allocated separate hardware resources. This is in contrast to parallelization, known as identical-program multi-data (SPMD), which involves dividing data across multiple identical copies of a model that exchange gradient information. Due to the relative lack of flexibility of SPMD, the latest generation of large models built to perform multiple heterogeneous tasks is designed based on MPMD. There are several advantages to using the SPMD architecture. Such advantages include, but are not limited to, (I) improved modularity, which allows different subsets of a model to be used for any given task without activating the entire model. This is particularly useful for fine-tuning a model across many different tasks. (II) the ability to efficiently utilize heterogeneous hardware.

[0049] Given the costs associated with training large-scale models and the abundant evidence that features learned to solve specific problems are useful for solving many others, such MPMD architectures, which enable greater modularity and flexibility, are likely to become increasingly important. Note that the approaches discussed below may at least compete with SPMD computations, which require the communication of gradients themselves between players.

[0050] When training with a batch of data, N corresponds to the product of the batch size and the input dimension. For classical data and models, the parameter region of interest is when N and P are large, and L remains moderate. For a typical unitary model, P l =O(N 2 ) This also scales the number of parameters in a fully connected network. {A l If L represents the encoding of some input data, then L is naturally (usually exponentially) related to the depth of the neural network with a polynomial activation function. For example, in the case of transformers, this depth is usually small. It is known to increase exponentially with the network depth across simple architectures suitable for expressiveness analysis, which helps explain why this domain is actually widely used. Some embodiments use the expressiveness of quantum circuits with similar structures. Such embodiments may use data-dependent unitaries acting on a single qubit.

[0051] Equation 2 may appear to be perfectly linear, and indeed it can represent many useful classes of classical neural networks. In particular, {A l} can be a complex nonlinear parameterization function of the data. Several examples are provided below.

[0052] (i.) Feedforward networks with polynomial activation functions: It is worth noting that, generally speaking, networks with polynomial nonlinearity are universal function approximators because they can use depth as a resource to compensate for the expressiveness gap between polynomial and non-polynomial functions.

[0053] (ii) Unitary transformers: Self-attention-based transformers form the backbone of large language models and more general foundational models. The self-attention layer, the central component of the transformer, is a weight matrix W Q , WK , W V ∈R N×N R as defined by S×N A map between sequences (where S is the sequence length) is given as follows:

number

number

[0054] Considering a simplified model that does not include the softmax operation and omits the normalization coefficient, the linear attention map is given as follows:

number

[0055] Iterating through this map K times (using a different weight matrix in each layer) yields a function of the following form:

number

[0056] (iii) Fine-tuning a pre-trained model: The simplest example of this is obtained by setting L=1. For input x, the features of the pre-trained model are given by B1×, while unitary A1 represents a fine-tuning layer that takes the pre-trained features as input. B1 can be kept constant while A1 is updated by gradient descent, or both layers can be updated simultaneously.

[0057] There is some evidence that specific types of features learned from data are localized to specific layers of deep models. For example, factual associations are thought to be stored in the central fully connected layer of the transformer. Therefore, some embodiments use a finer-grained scheme with parameterized layers in which different players encode different types of useful information.

[0058] Some embodiments use a representation model of the following form:

number

[0059] Such models include matrix A l (x) and B l Unitary constraints based on quantum mechanics may be imposed on the data. The imposition of such unitary constraints may depend on the encoding and reading methods of the selected data. For example, by mapping bits one-to-one with qubits and using additional space qubits, a reversible classical function of any data can be encoded in a unitary matrix A(x).

[0060] However, this could negate any spatial advantages, and no practical computational advantages can be obtained without additional processing. As described above, other embodiments use more general functions in amplitude and phase space, enabling exponential compression of data into quantum states. Such settings come with a cost that depends on the type of function, but A l It has been shown that (x) can represent almost any function of the data x. Some embodiments take a different approach to this question, conceived by work on the expressiveness of a particular class of neural networks, based on quantifying the strength of the correlation between parts of the input that the network can efficiently represent. As an example of a matrix, separation rank (denoted as sep) may be used. For a function f of two variables y and z, it is defined as follows:

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[0061] For example, if f cannot represent either the correlation between y and z, then sep(f)=1. When calculated for a particular class of neural networks, y and z are treated as subsets of higher-dimensional inputs. Separation rank can be used to quantify, for example, the induction bias of convolutional networks on learning local correlations, the effect of depth on recurrent networks, and the ability of transformers to capture correlations across sequences as a function of their depth and width.

[0062] For a linear transformer described in the form of Equation 6, the separation rank is known to depend exponentially with respect to the depth of the constitutive circuit for almost all values ​​of inputs and parameters, up to several thresholds set by the network width. Therefore, in the quantum setting, it is natural to similarly infer log(sep(φ;I,J))=Θ(L). General A lWhile this may seem self-evident at first glance, it requires specialization. One drawback of this type of result is that it is not robust, and therefore requires reinforcement to address measurement errors or finite precision.

[0063] Through the analysis of the Fourier spectra of such models, we examine the expressiveness of circuits with parameterized unitaries and interleaved data-dependent unitaries. Asymptotic universality results can be achieved for shallow models of this form, depending on the use of expressive data coding. However, the explicit class of Hamiltonians they consider has a qubit count that scales exponentially with the model depth. More general results can be obtained. (Note the relevance to QSP, where the parameterization matrix is ​​a single-qubit rotation and the expressiveness increases only linearly with depth).

[0064] Given two scalar variables x and y, we define it as follows:

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[0065] Lemma 1. {λ li {B} is extracted from an arbitrary continuous distribution according to independent and identical distributions (iid), and parameterized unitary {B} l}, where the real and imaginary parts of each matrix element are R PLLet it be a real analytical function of parameter Θ extracted from a subset of λ. li} and {B l Except for the set of measure 0s for the selection of}, the following holds: i) The number of non-zero Fourier components in L1

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[0066] This nearly saturates the upper bound of the number of frequencies that can be represented by various forms of circuits. This result can be trivially generalized to a setting in which y and z are replaced by a scalar / vector sequence. The separation rank means that complex correlations between different parts of a sequence can, in principle, be represented by such circuits. {B l The constraints on} are very relatively loose and apply to the selection of standard parameterized unitaries.

[0067] The main drawback of results like those in Lemma 1 is that they are not robust to measurement errors because they are based on constructing a state that is a superposition of exponentially numbered terms with equal weights. It is best to view this as a necessary but insufficient condition to ensure that this class of models is highly expressive. It is also natural to explore whether it is possible to easily represent any single combination of features from different layers in a way that is robust to measurement errors. In this way, O(N) L It can be shown that it is actually easy to represent ) different frequencies.

[0068] An intriguing question that may require a more advanced approach is whether these types of circuits can function as efficient universal function approximators. Given by Equation 9 {A l An L-layer circuit with {B} is a trainable unitary {B}l Maximum O(N) 2 Since it has independent degrees of freedom L), the O(N) suggested by Lemma 1 is only true when L = O(logN). 2L-1 This makes it possible to independently control the Fourier coefficients of ). Similar scaling has been observed in classical models such as linear transformers. In this context, it may be useful to consider other feature maps, including parameterized ones.

[0069] Considering the input to Equation 1, the embodiment attempts to address two problems. Problem 1 (Variable Inference). Alice and Bob each calculate an estimate of <φ|Z0|φ> within the additive error ε. Problem 2 (Variance gradient estimation). Alice is ∇ A Bob estimates the value of <φ|Z0|φ> as follows: B The estimated values ​​of <φ|Z0|φ> are given by L respectively. ∞ The calculation is performed within the range of the additive error ε.

[0070] Lemma 2. Problem 1 is O(L / ε 2 This can be solved by communicating O(logN) qubits over rounds.

[0071] Problem 2 is O(L 2 ) Throughout the rounds,

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[0072] By using direct reductions from known problems in communication complexity, it can be shown that many classical communications are necessary to solve the above problem. Further questions may arise regarding the strictness of the quantum algorithm with respect to the total number of qubits and rounds communicated. It has been shown that at least Ω(L) rounds may be required to obtain the exponential advantages of communication.

[0073] Lemma 3. The classical communication complexity of Problems 1 and 2 is Ω(max(√N,L)). The quantum (classical) protocol of either Problem 1 or Problem 2 is Ω(L) round communication or Ω(N / L) 4 It requires communication of one of ) qubits (bits).

[0074] The lower bound requires the ability to apply a general orthogonal matrix, which should be noted as meaning a circuit of depth poly(N). However, this unitary does not need to be parameterizable or trainable. We can then ask whether the lower bound can be strengthened to apply to a more restricted class of unitaries. Combining these two results, in the L=O(polylog(N)) domain, we gain an exponential advantage in communication complexity at the cost of the polynomial overhead in computational complexity.

[0075] Since there is a boundary in the complexity of the final output, whether it is a low-dimensional output of correlated elements or a model up to a few finite errors, quantum advantages are possible in these problems. This might lead one to think that encoding the data into the amplitude of a quantum state trivially yields an exponential advantage in communication complexity whenever the output takes such a form. However, by considering the problem of inference with a linear model, it is shown that the situation is slightly different.

[0076] Lemma 4. In the case of a distributed linear classification problem, any quantum advantage in communication is at most a polynomial.

[0077] The problem of estimating a single gradient vector is discussed above. To efficiently solve the optimization problem, it is natural to consider when these or other gradient estimators can be used. Other algorithms that require a single quantum measurement per iteration are shown in Figure 4A. Figure 4A shows an unrestricted example of the first algorithm 400, a quantum tomography algorithm, in various embodiments. More specifically, the first algorithm 400 deals with shadow tomography distributed gradient descent. A second intermediate output is determined at a third quantum node. Determining the second intermediate result is based on coding the second intermediate output and the quantum tomography algorithm. The result of the distributed computation is computed at a third quantum node. Computing the result of the distributed computation is based on the combination of the first and second intermediate outputs. The distributed computation involves providing the model with a first input and a second input.

[0078] The following algorithm can not only efficiently solve certain convex optimization problems, but can also be applied to fine-tune shadowmography.

[0079] The following is a generalization of the algorithm in distributed configuration. A l Consider the case where is the product of all rotations of l, that is,

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[0080] Figure 4B shows a non-restrictive second algorithm 420 for distributed stochastic coordinate descent in various embodiments. Given a distributed state of the above form, the second algorithm 420, or a similar algorithm, can be used to perform the optimization of Θ. A Hermitian measurement is defined as follows:

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[0081] To demonstrate convergence, it is sufficient to note that, by construction, ||g||2 = ||β||1 holds. With an appropriately selected step size, algorithm 2 for a convex function L

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[0082] this is,

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[0083] The time complexity of a single iteration of this algorithm is proportional to the forward path, thus matching the scaling of classical backpropagation. This is in contrast to the polynomial overhead of shadow tomography (see, e.g., Theorem 1). It requires only a single measurement and minimal additional overhead per iteration, suggesting that implementation on noisy, short-term devices may be feasible.

[0084] Consider a model with depth L, A L Assuming only the parameter is defined, this model is |φ f This is shown by >. The circuit leading to that unitary can contain multiple data-dependent unitaries, and thus can represent complex features of the data. Training only the final layer in this way is one method used by various embodiments to fine-tune a pre-trained model. In this setup,

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[0085] In the shadow tomography problem, the observables can be selected online and adaptively based on past measurements. Therefore, the stream of measurement operators can be easily defined by measuring P observables to estimate the gradient with respect to the initial parameter set, updating these parameters using gradient descent with step size η, and defining a new set of observables based on the updated parameters. Repeating this T times yields a total of PT observables. With the scaling in Lemma 2, the total amount of communication required over O(L) rounds is:

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[0086] Under appropriately selected step sizes, if L is a convex function,

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[0087] Gradient descent method

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[0088] The results of the above discussion depended on the so-called sublinear convergence rate of the general stochastic optimization of convex functions. Using an additional structure, linear convergence (with error r) can be achieved using the stochastic gradient. T We can obtain a value that scales in the form of (meaning r<1). This is possible, for example, when the cause of stochasticity is subsampling or when we have occasional access to a noise-free gradient to implement a variance reduction scheme, but neither seems applicable to the current setup. It is an interesting unsolved question to see if there is a way to obtain a linear convergence rate using a novel algorithm by leveraging the structure of quantum circuits. Apart from the advantage in time complexity, this may mean an exponential advantage in communication for a more general class of circuits.

[0089] Conversely, given only black-box access to a noisy gradient oracle, assuming a T-oracle query, it is known that linear convergence is impossible without additional structure, even for a strongly convex objective function, if an information-theoretic lower bound Ω(1 / T) is preserved for the error. These results can be used to demonstrate the optimality of algorithms that rely on a standard variational circuit optimization paradigm, which involves performing quantum measurements at each iteration and using them to update parameters. This implies that linear convergence is possible if the entire optimization process is performed coherently.

[0090] In addition to its advantage in communication complexity, the quantum algorithm outlined above has an inherent advantage from a privacy standpoint. It is known that the number of bits of information that can be extracted from an unknown quantum state is proportional to the number of qubits. Therefore, since the above algorithm requires exchanging a logarithmic number of state copies over O(logN) qubits, even if all communication between two players is intercepted, an attacker cannot extract information about classical input data or model parameters exceeding a logarithmic number of bits. Specifically,

[0091] System 1. If all communications between Alice and Bob are intercepted by the attacker, the attacker O ~ (L 2 Log 2 N-channel 4 Plog(L / δ) / ε 4 It is not possible to extract classical information exceeding ) bits.

[0092] It would also be interesting to explore how much information Bob or Alice can extract about the input of other players by executing the protocol. If this amount is also logarithmic, it provides additional privacy for both the model owner and the data owner. This would allow two parties who do not necessarily trust each other or the channels through which they communicate to collaborate in jointly training a distributed model or using one of the distributed models for inference, while only showing the missing parts of the information they hold.

[0093] It is also worth mentioning that data privacy is guaranteed in scenarios where the user holding the data also specifies the processing performed on the data. In this setup, Alice holds both data x and a complete description of the unitary she wants to apply to her own state. She can send these classical descriptions of the unitary to Bob, and as long as the data and features are transmitted in the form of quantum states, he can extract only a logarithmic amount of information about them. In this setup, there is naturally no advantage in terms of communication complexity, as the classical descriptions of the unitary scale like poly(N,P).

[0094] method Figure 5 shows a flowchart of exemplary method 500 for performing distributed computation via a model and a distributed quantum computing system (DQCS) in various embodiments. The DQCS includes a set of quantum nodes and a quantum communication network that connects the quantum nodes of the set of quantum nodes in a communicative manner. In block 502, a first quantum node of the set of quantum nodes receives a first input to the model. In block 504, the first quantum node performs a first quantum computation based on the first input and a first component of the model implemented by the first node. Performing the first quantum computation includes generating an encoding of a first intermediate output of the model. In block 506, a second quantum node of the set of nodes receives a second input to the model. In block 508, the second quantum node performs a second quantum computation based on a second input and a second component of the model implemented by the second quantum node. Performing the second quantum computation includes generating an encoding of a second intermediate output of the model. In block 510, the encoding of the first intermediate output is transmitted from the first quantum node to the third quantum node of the quantum node set via the quantum communication network. In block 512, the encoding of the second intermediate output is transmitted from the second quantum node to the third quantum node via the quantum communication network. In block 514, the first intermediate output is determined at the third quantum computing node. Determining the first intermediate output is based on the encoding of the first intermediate output and the quantum tomography algorithm. In block 516, the second intermediate output is determined at the third quantum node. Determining the second intermediate output is based on the encoding of the second intermediate output and the quantum tomography algorithm. In block 518, the result of the distributed computation is determined at the third quantum node. Determining the result of the distributed computation is based on the combination of the first and second intermediate outputs. The distributed computation includes providing the model with a first input and a second input.

[0095] In various embodiments, the model is a machine learning model that implements a neural network. The first and second inputs may be included in the model's training dataset. The results of the distributed computation may include the gradient of a loss function evaluated based on the first and second inputs provided to the model. The method may further include performing a gradient descent of the loss function at a third quantum node. The model's parameter set can be updated based on the gradient descent of the loss function. In some embodiments, updating the model's parameter set includes training the model. In other embodiments, updating the model's parameter set includes fine-tuning the model. The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0096] In some embodiments, performing a first quantum computation includes preparing a first quantum state of a first qubitset of a first quantum node, thereby storing the encoding of the first intermediate output of the model. Performing a second quantum computation may include preparing a second quantum state of a second qubitset of a second quantum node, thereby storing the encoding of the second intermediate output of the model.

[0097] In such embodiments, transmitting the coding of the first intermediate output includes transmitting the first quantum state of the first qubit set from the first quantum node to the third quantum node via the quantum communication network. Transmitting the coding of the second intermediate output may include transmitting the second quantum state of the second qubit set from the second quantum node to the third quantum node via the quantum communication network.

[0098] In these embodiments, transmitting a first quantum state of a first qubitset includes transmitting multiple copies of the first quantum state from a first quantum node to a third quantum node via a quantum communication network. Transmitting a second quantum state of a first qubitset may include transmitting multiple copies of the second quantum state from a second quantum node to a third quantum node via a quantum communication network. Determining a first intermediate output of the model may include providing multiple copies of the first quantum state to a quantum tomography algorithm. Determining a second intermediate output may include providing multiple copies of the second quantum state to a quantum tomography algorithm. The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0099] In various embodiments, the quantum communication network transmits quantum states via a quantum teleportation algorithm. In other embodiments, the quantum communication network transmits quantum states via photons that encode the quantum states. The model may be a pre-trained machine learning model. In such embodiments, the results of distributed computation include an inference time prediction for the pre-trained machine learning model based on a first input.

[0100] Other embodiments relate to a method that includes receiving a first input in a distributed quantum computing system (DQCS). The DQCS may have a plurality of quantum nodes and a quantum communication network. The first quantum computation is performed in a first quantum node among the plurality of nodes. The first quantum computation may be based on a first input and a model. The output of the first quantum computation may be communicated to a second quantum node among the plurality of quantum nodes via the quantum communication network. The results of the distributed computation are determined and / or computed in the second quantum node. Determining the results of the distributed computation may be based on the output of the first quantum computation and a quantum tomography algorithm performed in the second quantum node. The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0101] Such a method may further include receiving a second input in the DQCS. The second quantum computation may be performed at a third quantum node among multiple nodes. Performing the second quantum computation may be done based on the second input and model. The output of the second quantum computation may be communicated to the second quantum node via a quantum communication network. The results of the distributed computation may be further determined based on the output of the second quantum computation and a quantum tomography algorithm.

[0102] Another method may involve performing distributed computation via a model and a distributed quantum computing system (DQCS). The DQCS may include a first quantum node, a second quantum node, and a first quantum communication channel that connects the first and second quantum nodes in a communicative manner. The method includes receiving a first input at the first quantum node. A first component of the model is implemented at the first quantum node. The encoding of the first input is transmitted from the first quantum node to the second quantum node via the first quantum communication channel. The first quantum computation is performed at the second quantum node. The execution of the first quantum computation is based on the encoding of the first input and a second component of the model implemented at the second quantum node. Executing the first quantum computation produces a first output. The encoding of the first output is transmitted from the second quantum node to the first quantum node. The second quantum computation is performed at the first quantum node. The execution of the second quantum computation is based on the encoding of the first output and the first component of the model, and the second output is generated by performing the second quantum computation.

[0103] Additional Embodiments One non-limiting embodiment includes a method, which includes receiving a first input in a distributed quantum computing system (DQCS), the DQCS having a plurality of quantum nodes and a quantum communication network. The method further includes performing a first quantum computation at a first quantum node among the plurality of nodes based on the first input and a model. The method further includes communicating the output of the first quantum computation to a second quantum node among the plurality of quantum nodes via the quantum communication network. The method further includes determining the result of the distributed computation based on the output of the first quantum computation and a quantum tomography algorithm performed at the second quantum node.

[0104] The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0105] The method may further include receiving a second input in the DQCS. The second quantum computation is performed on a third quantum node among multiple nodes, based on the second input and model. The output of the second quantum computation is communicated to the second quantum node via a quantum communication network. Determining the results of the distributed computation is further based on the output of the second quantum computation and a quantum tomography algorithm.

[0106] The model may be a neural network implementing a machine learning model. The first and second inputs may be included in the model's training dataset. The distributed computation results may include the gradient of the loss function evaluated based on the first and second inputs provided to the model.

[0107] The method may further involve performing a gradient descent of the loss function at a second quantum node. The model's parameter set is updated based on the gradient descent of the loss function.

[0108] In some embodiments, updating the model's parameter set may include training the model.

[0109] In some embodiments, updating the model's parameter set includes fine-tuning the model.

[0110] Performing a first quantum computation may involve preparing a first quantum state of a first qubit set of a first quantum node, thereby storing the encoding of the output of the first quantum computation.

[0111] Communicating the output of the first quantum computation may include communicating the first quantum state of the first qubit set from the first quantum node to the second quantum node via a quantum communication network.

[0112] A quantum communication network can transmit quantum states via at least one of either a quantum teleportation algorithm or photons that encode quantum states.

[0113] Another method involves performing distributed computation via a model and a distributed quantum computing system (DQCS). The DQCS includes a set of quantum nodes and a quantum communication network that connects the quantum nodes of the set of quantum nodes in a communicative manner. The method includes receiving a first input to the model at a first quantum node of the set of quantum nodes. The method includes performing a first quantum computation at the first quantum node based on the first input and a first component of the model implemented by the first node, and performing the first quantum computation includes generating an encoding of a first intermediate output of the model. The method includes receiving a second input to the model at a second quantum node of the set of quantum nodes. The method includes performing a second quantum computation at the second quantum node based on a second input and a second component of the model implemented by the second quantum node, and performing the second quantum computation includes generating an encoding of a second intermediate output of the model. The method includes transmitting the encoding of the first intermediate output from the first quantum node to a third quantum node of the set of quantum nodes via the quantum communication network. The method includes transmitting the encoding of a second intermediate output from a second quantum node to a third quantum node via a quantum communication network. The method includes determining a first intermediate output at the third quantum node based on the encoding of the first intermediate output and a quantum tomography algorithm. The method includes determining a second intermediate output at the third quantum node based on the encoding of the second intermediate output and a quantum tomography algorithm. The method includes computing the results of a distributed computation at the third quantum node based on the combination of the first and second intermediate outputs, the distributed computation including providing the first and second inputs to a model.

[0114] The model may be a machine learning model with a neural network implemented, where the first and second inputs are included in the model's training dataset, and the distributed computation results include the gradient of a loss function evaluated based on the first and second inputs provided to the model.

[0115] The method may involve performing a gradient descent of the loss function at a third quantum node. The model's parameter set is updated based on the gradient descent of the loss function.

[0116] Updating the model's parameter set may include training the model.

[0117] Updating the model's parameter set may involve fine-tuning the model.

[0118] The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0119] Performing the first quantum computation involves preparing the first quantum state of the first qubitset of the first quantum node, thereby storing the encoding of the first intermediate output of the model; and performing the second quantum computation involves preparing the second quantum state of the second qubitset of the second quantum node, thereby storing the encoding of the second intermediate output of the model.

[0120] Transmitting a coding of the first intermediate output may include transmitting a first quantum state of the first qubit set from the first quantum node to the third quantum node via the quantum communication network, and transmitting a coding of the second intermediate output may include transmitting a second quantum state of the second qubit set from the second quantum node to the third quantum node via the quantum communication network.

[0121] Transmitting a first quantum state of a first qubitset may include transmitting multiple copies of the first quantum state from a first quantum node to a third quantum node via a quantum communication network, and transmitting a second quantum state of a first qubitset may include transmitting multiple copies of the second quantum state from a second quantum node to a third quantum node via a quantum communication network.

[0122] Determining the first intermediate output of the model may involve providing multiple copies of the first quantum state to a quantum tomography algorithm, and determining the second intermediate output may involve providing multiple copies of the second quantum state to a quantum tomography algorithm.

[0123] The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0124] In some embodiments, the quantum communication network may transmit quantum states via a quantum teleportation algorithm.

[0125] In some embodiments, the quantum communication network transmits quantum states via photons that encode the quantum states.

[0126] The model may be a pre-trained machine learning model, and the distributed computation results include an inference time prediction for the pre-trained machine learning model based on a first input.

[0127] Another method is for performing distributed computation via a distributed quantum computing system (DQCS) which includes a model and a first quantum node, a second quantum node, and a first quantum communication channel that connects the first and second quantum nodes in a communicative manner. The method includes receiving a first input at the first quantum node, and a first component of the model is implemented at the first quantum node. The method includes transmitting an encoding of the first input from the first quantum node to the second quantum node via the first quantum communication channel. The method includes performing a first quantum computation at the second quantum node based on the encoding of the first input and a second component of the model located at the second quantum node, and performing the first quantum computation produces a first output. The method includes transmitting an encoding of the first output from the second quantum node to the first quantum node via the first quantum communication channel. The method involves performing a second quantum computation at a first quantum node based on the encoding of the first output and the first component of the model, the second output being produced by performing the second quantum computation.

[0128] The DQCS may further include a third quantum node and a second quantum communication channel that connects the second and third quantum nodes in a communicative manner. The method may further include receiving a first input at the first quantum node, where the first component of the model is implemented at the first quantum node. The method further includes transmitting an encoding of the first input from the first quantum node to the second quantum node via the first quantum communication channel. The method includes performing a first quantum computation at the second quantum node based on the encoding of the first input and the second component of the model located at the second quantum node, thereby producing a first output. The method includes transmitting an encoding of the first output from the second quantum node to the first quantum node via the first quantum communication channel. The method includes performing a second quantum computation at the first quantum node based on the encoding of the first output and the first component of the model, thereby producing a second output.

[0129] One embodiment includes a distributed quantum computing system (DQCS). The DQCS includes a set of quantum nodes, including a first quantum node and a second quantum node. The DQCS includes a quantum communication network that connects the quantum nodes of the set of quantum nodes in a communicative manner. The DQCS includes one or more processors. The DQCS includes one or more memory devices, which, when executed by one or more processors, store computer-readable instructions that cause one or more processors to perform an operation for performing a distributed computation through a model. The operation includes receiving a first input to the model at the first quantum node of the set of quantum nodes. The operation includes performing a first quantum computation at the first quantum node based on the first input and a first component of the model implemented by the first node, and performing the first quantum computation includes generating an encoding of a first intermediate output of the model. The operation includes receiving a second input to the model at the second quantum node of the set of quantum nodes. The operation includes performing a second quantum computation at the second quantum node based on a second input and a second component of the model implemented by the second quantum node, the execution of which includes generating an encoding of the second intermediate output of the model. The operation includes transmitting the encoding of the first intermediate output from the first quantum node to the third quantum node of the quantum node set via the quantum communication network. The operation includes transmitting the encoding of the second intermediate output from the second quantum node to the third quantum node via the quantum communication network. The operation includes determining the first intermediate output at the third quantum node based on the encoding of the first intermediate output and a quantum tomography algorithm. The operation includes determining the second intermediate output at the third quantum node based on the encoding of the second intermediate output and a quantum tomography algorithm. The operation involves a third quantum node computing the results of a distributed computation based on a combination of the first and second intermediate outputs, and the distributed computation involves providing the first and second inputs to a model.

[0130] The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0131] One method includes one or more embodiments of any of the embodiments provided herein, or variations thereof.

[0132] Other embodiments include distributed quantum computing systems (DQCS) that include one or more embodiments or variations of any of the embodiments described herein.

[0133] One exemplary aspect of this disclosure relates to a method for performing a distributed computation. The distributed computation is performed via a model and a distributed quantum computing system (DQCS). The DQCS includes a set of quantum nodes and a quantum communication network that connects the quantum nodes of the set of quantum nodes in a communicative manner. The method includes receiving a first input to the model at a first quantum node of the set of quantum nodes. The first quantum computation is performed at the first quantum node. The execution of the first quantum computation is based on the first input and a first component of the model implemented by the first node. Performing the first quantum computation includes generating a coding of a first intermediate output of the model. A second input to the model is received at a second quantum node of the set of quantum nodes. The second quantum computation is performed at the second quantum node. The execution of the second quantum computation is based on a second input and a second component of the model implemented by the second quantum node. Performing the second quantum computation includes generating a coding of a second intermediate output of the model. The encoding of the first intermediate output is transmitted from the first quantum node to the third quantum node of the quantum node set via the quantum communication network. The encoding of the second intermediate output is transmitted from the second quantum node to the third quantum node via the quantum communication network. The first intermediate output is determined at the third quantum node. Determining the third quantum node is based on the encoding of the first intermediate output and the quantum tomography algorithm.

[0134] One exemplary aspect of this disclosure relates to a method for performing a distributed computation. The distributed computation is performed via a model and a distributed quantum computing system (DQCS). The DQCS includes a set of quantum nodes and a quantum communication network that connects the quantum nodes of the set of quantum nodes in a communicative manner. The method includes receiving a first input to the model at a first quantum node of the set of quantum nodes. The first quantum computation is performed at the first quantum node. The execution of the first quantum computation is based on the first input and a first component of the model implemented by the first node. Performing the first quantum computation includes generating a coding of a first intermediate output of the model. A second input to the model is received at a second quantum node of the set of quantum nodes. The second quantum computation is performed at the second quantum node. The execution of the second quantum computation is based on a second input and a second component of the model implemented by the second quantum node. Performing the second quantum computation includes generating a coding of a second intermediate output of the model. The encoding of the first intermediate output is transmitted from the first quantum node to the third quantum node of the quantum node set via the quantum communication network. The encoding of the second intermediate output is transmitted from the second quantum node to the third quantum node via the quantum communication network. The first intermediate output is determined at the third quantum node. Determining the third quantum node is based on the encoding of the first intermediate output and the quantum tomography algorithm. The second intermediate output is determined at the third quantum node. Determining the second intermediate output is based on the encoding of the second intermediate output and the quantum tomography algorithm. The result of the distributed computation is computed at the third quantum node. Computing the result of the distributed computation is based on the combination of the first and second intermediate outputs. The distributed computation includes providing the model with a first input and a second input.

[0135] The digital, classical, and / or quantum subjects, as well as embodiments of digital functional and quantum computations, described herein can be implemented in digital electronic circuits, appropriate quantum circuits, or, more generally, in quantum computing systems, in tangibly implemented digital and / or quantum computer software or firmware, in digital and / or quantum computing hardware including the structures disclosed herein and their equivalents, or in one or more combinations thereof. The term “quantum computing system” may include, but is not limited to, quantum computers / computing systems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0136] Embodiments of the digital and / or quantum subject matter described herein can be implemented as one or more digital and / or quantum computer programs, i.e., as one or more modules of digital and / or quantum computer program instructions encoded on a tangible, non-temporary storage medium for execution by a data processing device or for controlling the operation of a data processing device. The digital and / or quantum computer storage medium may be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubit / qubit structures, or one or more combinations thereof. Alternatively or additionally, the program instructions may be encoded into artificially generated propagating signals (e.g., machine-generated electrical, optical, or electromagnetic signals) capable of encoding digital and / or quantum information, such propagating signals are generated to encode the digital and / or quantum information for transmission to a suitable receiving device for execution by a data processing device.

[0137] The terms quantum information and quantum data refer to information or data transmitted by and held or stored by quantum systems, the smallest non-trivial system being a qubit, i.e., a system that defines the unit of quantum information. It will be understood that the term “qubit” encompasses all quantum systems that can be appropriately approximated as two-level systems in the corresponding context. Such quantum systems may include, for example, multi-level systems having two or more levels. Examples of such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many embodiments, the computational ground state is identified by the ground state and the first excited state, but it will be understood that other configurations are possible in which the computational state is identified by a higher-level excited state (e.g., a qubit).

[0138] The term “data processing device” refers to digital and / or quantum data processing hardware and encompasses all kinds of devices, machines, and equipment for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, or multiple digital and quantum processors or computers, and combinations thereof. A device may also be, or further include, a dedicated logic circuit, e.g., an FPGA (Field-Programmable Gate Array), or an ASIC (Application-Specific Integrated Circuit), or a quantum simulator, i.e., a quantum data processing device designed to simulate or generate information about a particular quantum system. Specifically, a quantum simulator is a dedicated quantum computer that does not have the ability to perform general-purpose quantum computations. A device may optionally include, in addition to hardware, code that constructs an execution environment for digital and / or quantum computer programs, e.g., processor firmware, protocol stacks, database management systems, operating systems, or one or more combinations thereof.

[0139] Digital or classical computer programs, which may also be called or described as programs, software, software applications, modules, software modules, scripts, or code, may be written in any form of programming language, including compiled languages ​​or interpreted languages, or declarative languages ​​or procedural languages, and such computer programs may be deployed in any form, including as standalone programs, or as modules, components, subroutines, or other units suitable for use in a digital computing environment. Quantum computer programs, which may also be called or described as programs, software, software applications, modules, software modules, scripts, or code, may be written in any form of programming language, including compiled languages ​​or interpreted languages, or declarative languages ​​or procedural languages, and converted into a suitable quantum programming language, or may be written in a quantum programming language, such as QCL, Quipper, or Cirq.

[0140] Digital and / or quantum computer programs may, but do not necessarily, correspond to files in a file system. A program may be stored in a single file dedicated to the program, or in a group of files, such as one or more scripts stored in a markup language document, or in a group of files containing modules, subprograms, or sections of code. Digital and / or quantum computer programs may be deployed to run on a single digital computer or a single quantum computer, or they may be located in one place or distributed across multiple locations and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network capable of transmitting quantum data using quantum systems, such as qubits. Generally, digital data communication networks cannot transmit quantum data, while quantum data communication networks can transmit both quantum and digital data.

[0141] The processes and logic flows described herein may be executed by one or more programmable digital and / or quantum computers operating together with one or more digital and / or quantum processors to execute one or more digital and / or quantum computer programs that function by performing operations on input digital and quantum data to generate outputs as needed. The processes and logic flows may also be executed by a combination of one or more programmed digital and / or quantum computers and dedicated logic circuits or quantum simulators, such as FPGAs or ASICs, and the apparatus may be implemented in this manner.

[0142] For a system consisting of one or more digital and / or quantum computers or processors to be “configured” or “operable” to perform a particular operation or action means that software, firmware, hardware, or a combination thereof is installed on the system that causes the system to perform that operation or action when it is in operation. For one or more digital and / or quantum computer programs to be configured to perform a particular operation or action means that one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform that operation or action. A quantum computer may receive instructions from a digital computer, and when executed by a quantum computing device, causes the device to perform an operation or action.

[0143] A digital and / or quantum computer suitable for executing digital and / or quantum computer programs may be based on a general-purpose or dedicated digital and / or quantum microprocessor, or both, or any other type of central digital and / or quantum processing unit. Generally, the central digital and / or quantum processing unit receives instructions and digital and / or quantum data from read-only memory, or random-access memory, or a quantum system suitable for transmitting quantum data such as photons, or a combination thereof.

[0144] Some exemplary elements of a digital and / or quantum computer are a central processing unit for executing or running instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and memory may be complemented by or incorporated into dedicated logic circuits or quantum simulators. Generally, a digital and / or quantum computer also includes one or more mass storage devices for storing digital and / or quantum data, such as magnetic, magneto-optical disks, or optical disks, or quantum systems suitable for storing quantum information, or is operablely coupled to them to receive digital and / or quantum data from them, transfer digital and / or quantum data to them, or both. However, a digital and / or quantum computer is not required to have such devices.

[0145] Suitable digital and / or quantum computer-readable media for storing digital and / or quantum computer program instructions and digital and / or quantum data include, for example, all forms of non-volatile digital and / or quantum memory, media and memory devices, and quantum systems (e.g., trapped atoms or electrons), including semiconductor memory devices (e.g., EPROM, EEPROM, and flash memory devices), magnetic disks (e.g., internal hard disks or removable disks), magneto-optical disks, and CD-ROM and DVD-ROM disks. Quantum memory is understood to be a device capable of storing quantum data for long periods with high fidelity and efficiency, such as an optical-matter interface that uses light for transmission and matter for storing and preserving quantum features of quantum data, such as superposition or quantum coherence.

[0146] Control of the various systems or parts thereof described herein may be implemented in a digital and / or quantum computer program product containing instructions, which are stored in one or more tangible, non-temporary, machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or parts thereof described herein may each be implemented as an apparatus, method, or electronic system which may include one or more digital and / or quantum processing devices and a memory for storing executable instructions for performing the operations described herein.

[0147] This specification includes details of many specific embodiments, but these should not be construed as limiting the scope of claims, but rather as descriptions of features that may be specific to a particular embodiment. Certain features described herein in the context of a separate embodiment may also be implemented in combination in a single embodiment. Conversely, various features described in the context of a single embodiment may also be implemented in multiple embodiments, separately or in any preferred partial combination. Furthermore, even if features are described above as functioning in a particular combination and are initially claimed as such, one or more features in the claimed combination may be removed from the combination in some cases, and the claimed combination may cover a partial combination or a variation of a partial combination.

[0148] Similarly, while the operations are shown in a specific order in the drawings, this should not be understood as meaning that such operations must be performed in a specific or sequential order shown, or that all the operations exemplified must be performed, in order to achieve the desired result. In certain situations, multitasking and parallel processing may be advantageous. Furthermore, the separation of various system modules and components in the embodiments described above should not be understood as requiring such separation in all embodiments, and the program components and systems described above should generally be understood as being able to be integrated into a single software product or packaged into multiple software products.

[0149] Specific embodiments of the subject matter have been described. Other embodiments are also within the scope of the following claims. For example, the actions enumerated in the claims can still achieve the desired results even if they are performed in a different order. As an example, the process depicted in the accompanying drawings does not necessarily require the specific order or sequence shown to achieve the desired results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. Receiving a first input in a distributed quantum computing system (DQCS) having multiple quantum nodes and a quantum communication network, Based on the first input and model, a first quantum computation is performed at the first quantum node among the plurality of quantum nodes, The output of the first quantum computation is communicated to a second quantum node among the plurality of quantum nodes via the quantum communication network, The results of the distributed computation are determined based on the output of the first quantum computation and the quantum tomography algorithm performed at the second quantum node, Methods that include...

2. The method according to claim 1, wherein the quantum tomography algorithm is a shadow quantum tomography algorithm.

3. The DQCS receives a second input, Based on the second input and the model, a second quantum computation is performed at the third quantum node among the plurality of quantum nodes, The output of the second quantum computation is communicated to the second quantum node via the quantum communication network, The result of the distributed computation is determined based on the output of the second quantum computation and the quantum tomography algorithm, The method according to claim 1, further comprising:

4. The method according to claim 3, wherein the model is a machine learning model in which a neural network is implemented, the first input and the second input are included in the training dataset of the model, and the result of the distributed computation includes the gradient of a loss function evaluated based on the first input and the second input provided to the model.

5. At the second quantum node, the gradient descent of the loss function is performed, The parameter set of the model is updated based on the descent of the gradient of the loss function, The method according to claim 4, further comprising:

6. The method according to claim 5, wherein updating the parameter set of the model is a means of training the model.

7. The method according to claim 5, wherein updating the parameter set of the model includes fine-tuning the model.

8. The method according to claim 1, wherein performing the first quantum computation comprises preparing a first quantum state of a first qubit set of the first quantum node, the first quantum state of the first qubit set storing an encoding of the output of the first quantum computation.

9. The method according to claim 8, wherein communicating the output of the first quantum computation includes communicating the first quantum state of the first qubit set from the first quantum node to the second quantum node via the quantum communication network.

10. The method according to claim 1, wherein the quantum communication network transmits a quantum state via at least one of a quantum teleportation algorithm or a photon encoding a quantum state.

11. A method for performing distributed computation via a model and a distributed quantum computing system (DQCS) which includes a set of quantum nodes and a quantum communication network that connects the quantum nodes of the set of quantum nodes in a communicative manner, The first quantum node of the quantum node set receives the first input to the model, Performing a first quantum computation at the first quantum node based on the first input and a first component of the model implemented by the first quantum node, the performance of the first quantum computation includes generating an encoding of a first intermediate output of the model. The second quantum node of the quantum node set receives the second input to the model, Performing a second quantum computation at the second quantum node based on the second input and the second component of the model implemented by the second quantum node, the performance of the second quantum computation includes generating an encoding of the second intermediate output of the model. The encoding of the first intermediate output is transmitted from the first quantum node to the third quantum node of the quantum node set via the quantum communication network, The encoding of the second intermediate output is transmitted from the second quantum node to the third quantum node via the quantum communication network, The third quantum node determines the first intermediate output based on the coding and quantum tomography algorithm of the first intermediate output, The third quantum node determines the second intermediate output based on the coding of the second intermediate output and the quantum tomography algorithm, A method comprising computing the result of the distributed computation at the third quantum node based on a combination of the first intermediate output and the second intermediate output, wherein the distributed computation includes providing the first input and the second input to the model.

12. The method according to claim 11, wherein the model is a machine learning model in which a neural network is implemented, the first input and the second input are included in the training dataset of the model, and the result of the distributed computation includes the gradient of a loss function evaluated based on the first input and the second input provided to the model.

13. At the third quantum node, the gradient descent of the loss function is performed, Further comprising updating the parameter set of the model based on the descent of the gradient of the loss function, The method according to claim 12.

14. The method according to claim 13, wherein updating the parameter set of the model is a means of training the model.

15. The method according to claim 13, wherein updating the parameter set of the model includes fine-tuning the model.

16. The method according to claim 11, wherein the quantum tomography algorithm is a shadow quantum tomography algorithm.

17. The method according to claim 11, wherein performing the first quantum computation comprises preparing a first quantum state of a first qubit set of the first quantum node, the first quantum state of the first qubit set storing the encoding of the first intermediate output of the model, and performing the second quantum computation comprises preparing a second quantum state of a second qubit set of the second quantum node, the second quantum state of the second qubit set storing the encoding of the second intermediate output of the model.

18. The method according to claim 17, wherein transmitting the coding of the first intermediate output comprises transmitting the first quantum state of the first qubit set from the first quantum node to the third quantum node via the quantum communication network, and transmitting the coding of the second intermediate output comprises transmitting the second quantum state of the second qubit set from the second quantum node to the third quantum node via the quantum communication network.

19. The method according to claim 18, wherein transmitting the first quantum state of the first qubit set comprises transmitting a plurality of copies of the first quantum state from the first quantum node to the third quantum node via the quantum communication network, and transmitting the second quantum state of the first qubit set comprises transmitting a plurality of copies of the second quantum state from the second quantum node to the third quantum node via the quantum communication network.

20. The method according to claim 19, wherein determining the first intermediate output of the model comprises providing the plurality of copies of the first quantum state to the quantum tomography algorithm, and determining the second intermediate output comprises providing the plurality of copies of the second quantum state to the quantum tomography algorithm.

21. The method according to claim 20, wherein the quantum tomography algorithm is a shadow quantum tomography algorithm.

22. The method according to claim 11, wherein the quantum communication network transmits quantum states via a quantum teleportation algorithm.

23. The method according to claim 11, wherein the quantum communication network transmits quantum states via photons that encode quantum states.

24. The method according to claim 11, wherein the model is a trained machine learning model, and the result of the distributed computation includes an inference time prediction of the trained machine learning model based on the first input.

25. A method for performing distributed computation via a model and a distributed quantum computing system (DQCS) including a first quantum node, a second quantum node, and a first quantum communication channel that connects the first quantum node and the second quantum node in a communicative manner, The first quantum node receives a first input, and the first component of the model is implemented at the first quantum node, which receives the input. The encoding of the first input is transmitted from the first quantum node to the second quantum node via the first quantum communication channel, Performing a first quantum computation at the second quantum node based on the encoding of the first input and the second component of the model located at the second quantum node, wherein the first quantum computation generates a first output. The encoding of the first output is transmitted from the second quantum node to the first quantum node via the first quantum communication channel, A method comprising performing a second quantum computation at the first quantum node based on the encoding of the first output and the first component of the model, wherein the second quantum computation generates a second output.

26. The DQCS further includes a third quantum node and a second quantum communication channel that connects the second quantum node and the third quantum node in a communicative manner. The third quantum node receives the second input, and the third component of the model is located at the third quantum node and receives the input. The encoding of the second input is transmitted from the third quantum node to the second quantum node via the second quantum communication channel, Performing a third quantum computation at the second quantum node based on the encoding of the second input and the second component of the model, wherein the third quantum computation generates a third output. The encoding of the third output is transmitted from the second quantum node to the third quantum node via the second quantum communication channel, The method according to claim 25, comprising performing a fourth quantum computation on the third quantum node based on the encoding of the third output and the third component of the model implemented by the third quantum node, wherein performing the fourth quantum computation generates a fifth output.

27. A distributed quantum computing system (DQCS), A quantum node set including a first quantum node and a second quantum node, A quantum communication network that connects the quantum nodes of the quantum node set in a communicative manner, One or more processors, A system comprising one or more memory devices, wherein the one or more memory devices, when executed by the one or more processors, store computer-readable instructions that cause the one or more processors to perform an operation for performing distributed computation via a model, and the operation is The first quantum node of the quantum node set receives the first input to the model, Performing a first quantum computation at the first quantum node based on the first input and a first component of the model implemented by the first quantum node, the performance of the first quantum computation includes generating an encoding of a first intermediate output of the model. The second quantum node of the quantum node set receives the second input to the model, Performing a second quantum computation at the second quantum node based on the second input and the second component of the model implemented by the second quantum node, the performance of the second quantum computation includes generating an encoding of the second intermediate output of the model. The encoding of the first intermediate output is transmitted from the first quantum node to the third quantum node of the quantum node set via the quantum communication network, The encoding of the second intermediate output is transmitted from the second quantum node to the third quantum node via the quantum communication network, The third quantum node determines the first intermediate output based on the coding and quantum tomography algorithm of the first intermediate output, The third quantum node determines the second intermediate output based on the coding of the second intermediate output and the quantum tomography algorithm, A distributed quantum computing system (DQCS) comprising: computing the result of the distributed computation at the third quantum node based on a combination of the first intermediate output and the second intermediate output, wherein the distributed computation includes providing the first input and the second input to the model.

28. The DQCS according to claim 27, wherein the quantum tomography algorithm is a shadow quantum tomography algorithm.

29. A method comprising one or more embodiments of any one of the embodiments or variations thereof described herein.

30. A distributed quantum computing system (DQCS) comprising one or more embodiments or variations thereof of any one of the embodiments described herein.

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