Distributed machine learning via quantum computation and quantum communication

Abstract

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 performed at a first quantum node of 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 of the plurality of quantum nodes over the quantum communication network. A result of a distributed computation based on the output of the first quantum computation and a quantum tomography algorithm implemented at the second quantum node.

Description

DISTRIBUTED MACHINE LEARNING VIA QUANTUM COMPUTATION AND QUANTUM COMMUNICATION PRIORITY

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 583,108, entitled DISTRIBUTED MACHINE LEARNING VIA QUANTUM COMPUTATION AND QUANTUM COMMUNICATION, filed on September 15, 2023, the contents of which are incorporated herein by reference in the entirety. FIELD

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

[0003] Quantum computing is a computing method that takes advantage of quantum effects, such as superposition of basis states and entanglement to perform certain computations more efficiently than a classical digital computer. In contrast to a digital computer, which stores and manipulates information in the form of bits, e.g., a “1” or “0,” quantum computing systems can manipulate information using quantum bits (“qubits”). A qubit can refer to a quantum device that provides the superposition of multiple states, e.g., data in both the “0” and “1” state, and / or to the superposition of data, itself, in the multiple states. In accordance with conventional terminology, the superposition of a “0” and “1” state in a quantum system may be represented, e.g., as a |0〉 + b |1〉 The “0” and “1” states of a digital computer are analogous to the |0〉 and |1〉 basis states, respectively of a qubit. SUMMARY

[0004] Aspects and advantages of embodiments of the present disclosure will be set forth in part in the following description, 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 performed at a first quantum node of the plurality of nodes based on the first input and a model. An output of the first quantumcomputation is communicated to a second quantum node of the plurality of quantum nodes over the quantum communication network. A result of a distributed computation based on the output of the first quantum computation and a quantum tomography algorithm implemented 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 become 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 example embodiments of the present disclosure and, together with the description, explain the related principles. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Detailed discussion of embodiments directed to one of ordinary skill in the art is set forth in the specification, which refers to the appended figures, in which:

[0009] FIG.1 depicts an example quantum computing system according to example embodiments of the present disclosure.

[0010] FIG.2A shows a non-limiting example of a distributed computing environment that the various embodiments may be applied within.

[0011] FIG.2B shows a schematic of the bandwidth requirements for both classical and quantum communication protocols, when computing a gradient, for various embodiments.

[0012] FIG.3 shows a schematic diagram for a distributed quantum circuit, according to various embodiments.

[0013] FIG.4A shows a non-limiting example of a first algorithm that is a quantum tomography algorithm, according to various embodiments.

[0014] FIG.4B shows a non-limiting second algorithm that is directed towards distributed probabilistic coordinate descent, according to various embodiments.

[0015] FIG.5 depicts a flow diagram of an example method for performing a distributed computation via a model and a distributed quantum computation system, according to various embodiments.DETAILED DESCRIPTION

[0016] Example aspects of the present disclosure are directed towards implementing machine learning (e.g., both training of models and the deployment of the trained models) via distributed computing. More particularly, the embodiments implement distributed quantum computing in the context of training and deploying machine-learned models. In the discussion throughout, the various machine-learned learned models may be implemented via one or more distributed neural network architectures. Although the embodiments are not so limited, and other distributed architectures may be employed to implement the machine-learned models of the various embodiments. Quantum communication protocols are implemented to configure quantum communication channels between the nodes in the distributed quantum computing assets. The quantum communication protocols leverage quantum shadow tomography, such that the number of qubits that have to be communicated between the quantum computing nodes is exponentially smaller than the number of parameters in the neural network and the size of the (training or runtime) inputs.

[0017] As the scale of the datasets and parameterized models used to perform computation over data continues to grow, distributing workloads across hardware becomes useful for enabling progress. This may require considering not only computational complexity, but also that communication may become the bottleneck in large-scale training and inference. While most often discussed in the context of computational advantages, quantum technology also offers exponential reductions in the amount data of that needs to be transferred for some tasks and these communication advantages could impact future distributed learning models dramatically. While neural network architectures continually evolve, one invariant feature in their use is the key role of gradient-based optimization when fitting these models to data. This typically requires communication of data, features and / or gradients between nodes that perform parameter updates. Using a model for inference on user-provided data also requires communication of either the model or the input, which can itself be of considerable size. Large pre-trained models are often fine-tuned to solve downstream tasks, which involves appending additional layers to the end of a pre-trained model and performing additional training. This requires communicating either the model itself or inputs and features. It is shown below that for a reasonably generic class of learning models, if quantum bits (aka qubits) are communicated instead of classical bits, an exponential reduction in the communication required to solve such problems can be achieved. This protocol additionally guarantees privacy of both the user data and model parameters through natural features of quantum mechanics, without the need for additional cryptographic or privacy protocols. These types ofcommunication advantages help scope the future roles and interplay between quantum and classical communication for distributed machine learning

[0018] More specifically, the size of neural networks and the datasets the neural networks are being trained on are continuously being scaled up, since scaling tends to improve performance. Beyond a certain point, training may be done in a distributed fashion on multiple machines / groups of machines. As noted above, this may require communication of data or internal features learned by the model between workers in order to train the model or to use it for inference once trained. By employing distributed quantum computing and quantum communication methods, the embodiments reduce the communication cost when training a distributed, parameterized model using gradient descent and when using a trained model for inference. Thus, the embodiments facilitate training on datasets that are too large to centrally store.

[0019] The embodiments may be used in any setting where classical computers are employed in the storing of a distributed neural network model and the training data. In the embodiments, the classical computers (e.g., distributed amongst a set of nodes) are replaced by quantum computers (e.g., distributed amongst the set of nodes). Furthermore, rather than classical networks connecting the nodes, in the embodiments, quantum communication channels communicatively couple the quantum computers of the set of nodes. The quantum channels are configured to transmit coherent quantum states between the nodes. In these settings, the embodiments reduce the task of computing the output of the network (e.g., when provided with some inputs), as well as the task of computing the gradients of the network parameters to quantum shadow tomography methods. The embodiments establish lower bounds on the number of copies of quantum states needed for shadow tomography and transmit that number of copies of a coherent quantum state between nodes. It is shown that the number of qubits that have to be communicated between the quantum computers is exponentially smaller than the number of parameters in the network and the size of the input.

[0020] The embodiments may be employed to train a large, distributed model using gradient descent. The embodiments may also be employed at inference time for a large model, e.g., when a large trained model is presented with user data. The embodiments may also be used for fine-tuning a trained model by appending additional layers. All these scenarios exhibit a compositional structure that will benefit from quantum communication, as presented herein.

[0021] In addition to reducing the communication complexity via quantum communication channels and quantum shadow tomography, the embodiments provide an additional advantage: data privacy. The data privacy advantage stems from the destructive nature of quantum measurement. This implies that the communication required to solve these problems is inherentlymore private than classical communication, since even if all the communication was intercepted by a third party, only a logarithmic amount of information can be extracted about the input data. This is useful both for guaranteeing the privacy of user data and the parameters of a trained model.

[0022] From determined lower bounds on the communication complexity of related problems, it is proven that at least in some settings, the amount of classical communication will have to be essentially proportional to the size of the input in order to solve this problem. The embodiments provide exponential savings in communication complexity compared to the standard methods of training and inference with neural networks.

[0023] 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 performed at a first quantum node of 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 of the plurality of quantum nodes over the quantum communication network. A result of a distributed computation based on the output of the first quantum computation and a quantum tomography algorithm implemented at the second quantum node.

[0024] Aspects of the present disclosure provide a number of technical effects and benefits. For instance, the embodiments configure the training and deployment of large-scale machined- learned models that are distributed across a plurality of quantum computing nodes. The quantum nodes communicate with one another using quantum shadow tomography. The employment of shadow tomography provides an exponential decrease in the amount of qubits that need to be transferred amongst the nodes, with respect to the size of the models and training data sets, thus providing a significant reduction in the communication complexity. This exponential prediction in communication complexity provides the distribution of large-scale models (e.g., for model training. Model fine-tuning, and inference time), as compared to classical methods. The embodiments also provide the advantage of data privacy, inherent in quantum communication channels.

[0025] FIG.1 depicts an example quantum computing system 100. The system 100 is an example of a system of one or more classical computers and / or quantum computing devices in one or more locations, in which the systems, components, and techniques described below can be implemented. Those of ordinary skill in the art, using the disclosures provided herein, will understand that other quantum computing devices or systems can be used without deviating from the scope of the present disclosure.

[0026] The system 100 includes quantum hardware 102 in data communication with one or more classical processors 104. The classical processors 104 can be configured to execute computer-readable instructions stored in one or more memory devices to perform operations, such as 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, control device(s) 112, and readout device(s) 114 (e.g., readout resonator(s)). The quantum system 110 can include one or more multi-level quantum subsystems, such as a register of qubits (e.g., qubits 120). In some implementations, the multi-level quantum subsystems can include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, spin-based qubits, and the like.

[0027] The type of multi-level quantum subsystems that the system 100 utilizes may vary. For example, in some cases it may be convenient to include one or more readout device(s) 114 attached to one or more superconducting qubits, e.g., transmon, flux, gmon, xmon, or other qubits. In other cases, ion traps, photonic devices or superconducting cavities (e.g., with which states may be prepared 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.

[0028] Quantum circuits may be constructed and applied to the register of qubits included in the quantum system 110 via multiple control lines that are coupled to one or more control devices 112. Example control devices 112 that operate on the register of qubits can be used to implement quantum gates or quantum circuits having a plurality of quantum gates, e.g., Pauli gates, Hadamard gates, controlled-NOT (CNOT) gates, controlled-phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, etc. The one or more control devices 112 may be configured to operate on the quantum system 110 through one or more respective control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystems may be superconducting qubits and the control devices 112 may be configured to provide control pulses to control lines to generate magnetic fields to adjust the frequency of the qubits.

[0029] The quantum hardware 102 may further include readout devices 114 (e.g., readout resonators). Measurement results 108 obtained via measurement devices may be provided to the classical processors 104 for processing and analyzing. In some implementations, the quantum hardware 102 may include a quantum circuit and the control device(s) 112 and readout devices(s) 114 may implement one or more quantum logic gates that operate on the quantum system 100 through physical control parameters (e.g., microwave pulses) that are sent through wires includedin the quantum hardware 102. Further examples of control devices include arbitrary waveform generators, wherein a DAC (digital to analog converter) creates the signal.

[0030] The readout device(s) 114 may be configured to perform quantum measurements on the quantum system 110 and send measurement results 108 to the classical processors 104. In addition, the quantum hardware 102 may be configured to receive data specifying physical control qubit parameter values 106 from the classical processors 104. The quantum hardware 102 may use the received physical control qubit parameter values 106 to update the action of the control device(s) 112 and readout devices(s) 114 on the quantum system 110. For example, the quantum hardware 102 may receive data specifying new values representing voltage strengths of one or more DACs included in the control devices 112 and may update the action of the DACs on the quantum system 110 accordingly. The classical processors 104 may be configured to initialize the quantum system 110 in an initial quantum state, e.g., by sending data to the quantum hardware 102 specifying an initial set of parameters 106.

[0031] In some implementations, the readout device(s) 114 can take advantage of a difference in the impedance for the |0〉 and |1〉 states of an element of the quantum system, such as a qubit, to measure the state of the element (e.g., the qubit). For example, the resonance frequency of a readout resonator can take on different values when a qubit is in the state |0〉 or the state |1〉, due to the nonlinearity of the qubit. Therefore, a microwave pulse reflected from the readout device 114 carries an amplitude and phase shift that depend on the qubit state. In some implementations, a Purcell filter can be used in conjunction with the readout device(s) 114 to impede microwave propagation at the qubit frequency.

[0032] In some embodiments, the quantum system 110 can include a plurality of qubits 120 arranged, for instance, in a two-dimensional grid 122. For clarity, the two-dimensional grid 122 depicted in FIG.1 includes 4x4 qubits, however in some implementations the system 110 may include a smaller or a larger number of qubits. In some embodiments, the multiple qubits 120 can interact with each other through multiple qubit couplers, e.g., qubit coupler 124. The qubit couplers can define nearest neighbor interactions between the multiple qubits 120. In some implementations, the strengths of the multiple qubit couplers are tunable parameters. In some cases, the multiple qubit couplers included in the quantum computing system 100 may be couplers with a fixed coupling strength.

[0033] In some implementations, the multiple qubits 120 may include data qubits, such as qubit 126 and measurement qubits, such as qubit 128. A data qubit is a qubit that participates in a computation being performed by the system 100. A measurement qubit is a qubit that may be used to determine an outcome of a computation performed by the data qubit. That is, during acomputation an unknown state of the data qubit is transferred to the measurement qubit using a suitable physical operation and measured via a suitable measurement operation performed on the measurement qubit.

[0034] In some implementations, each qubit in the multiple qubits 120 can be operated using respective operating frequencies, such as an idling frequency and / or an interaction frequency and / or readout frequency and / or reset frequency. The operating frequencies can vary from qubit to qubit. For instance, each qubit may idle at a different operating frequency. The operating frequencies for the qubits 120 can be chosen before a computation is performed.

[0035] FIG.1 depicts one example quantum computing system that can be used to implement the methods and operations according to example aspects of the present disclosure. Other quantum computing systems can be used without deviating from the scope of the present disclosure.

[0036] Training and inference with large machine learning models that far exceed the memory capacity of individual devices may necessitate the design of distributed architectures, forcing one to contend with communication constraints. The embodiments include a framework for distributed computation over a quantum network in which data is encoded into specialized quantum states. The embodiments provide machine learning (ML) quantum models within this framework. Training for these models uses gradient descent. Within the quantum framework of the embodiments, both inference and gradient descent for these models can be performed with exponentially less communication compared to their classical analogs, and with relatively modest overhead relative to standard (classical) gradient-based methods. Some graph neural networks are particularly amenable to implementation within this framework. That is, the embodiments provide an exponential quantum advantage for a generic class of machine learning problems that hold regardless of the data encoding cost. Moreover, models in this class can encode highly nonlinear features of their inputs, and their expressivity increases exponentially with model depth. Due to the inherent nature of quantum states, the embodiments provide natural privacy advantages in the communicated quantum states that limit the amount of information that can be extracted from them about the data and model parameters. The embodiments provide foundation for distributed machine learning over quantum networks.

[0037] As the scale of the datasets and parameterized models used to perform computation over data continues to grow, distributing workloads across multiple devices may be useful for enabling progress. The choice of architecture for large-scale training and inference must not only make the best use of computational and memory resources, but also contend with the fact that communication may become a bottleneck. This is particularly pertinent as models grow so largethat they cannot rely on high-bandwidth interconnects within datacenters but are instead trained across multiple datacenters. When using modern optical interconnects, classical computers exchange bits represented by light. This however does not fully utilize the potential of the physical substrate; given suitable computational capabilities and algorithms, the quantum nature of light can be harnessed as a powerful communication resource. The embodiments provide a broad class of parameterized models, if quantum bits (qubits) are communicated instead of classical bits, an exponential reduction in the communication required to perform inference and gradient-based training can be achieved. This protocol additionally guarantees improved privacy of both the user data and model parameters through natural features of quantum mechanics, without the need for additional cryptographic or privacy protocols. The embodiments provide a general exponential quantum advantage on problems that occur naturally in the training and deployment of large machine learning models. These types of communication advantages help scope the future roles and interplay between quantum and classical communication for distributed machine learning.

[0038] Quantum computers may provide dramatic speedups across a number of computational tasks. One prominent example of these tasks being the ability to revolutionize an understanding of nature by enabling the simulation of quantum systems, owing to the natural similarity between quantum computers and the world. However, much of the data that one would like to compute within practice seems to come from an emergent classical world rather than directly exhibiting quantum properties. While there are some known examples of exponential quantum speedups for classical problems, most famously factoring and related hidden subgroup problems, these tend to be isolated and at times difficult to relate to practical applications that involve learning from data. In addition, even though significant speedups are known for certain ubiquitous problems in machine learning such as matrix inversion and principal component analysis, the advantage is often lost when including the cost of loading classical data into the quantum computer or of reading out the result into classical memory. This is because the complexity of loading dense classical data into the amplitudes of a quantum state (which is typically the encoding needed to obtain an exponential runtime advantage) and of reading out the amplitudes from a quantum state into classical memory, are both polynomial in the number of amplitudes. In applications where an efficient data access model avoids the above pitfalls, the complexity of quantum algorithms tends to depend on condition numbers of matrices which scale with system size in a way that reduces or even eliminates any quantum advantage. It is worth noting that much of the discussion about the impact of quantum technology on machine learning has focused on computational advantage. However, quantum resources are not only useful in reducing computational complexity — they can also provide an advantage in communicationcomplexity, enabling exponential reductions in communication for some problems. The embodiments leverage the advantages in quantum communication across a wide class of machine learning models. This advantage holds without requiring any sparsity assumptions or elaborate data access models.

[0039] FIG.2A shows a non-limiting example of distributed computing environment 200 that the various embodiments may be applied within. The distributed computing environment 200 includes multiple sets of computing resources that are distributed across space. Without loss of generality, distributed computing environment 200 includes a first set of resources 210, a second set of resources 212, a third set of resources 214, and a fourth set of resources 216. A set of resources may be referred to as a node of resources (or simply as a node). The various sets of resources may include classical and / or quantum computational resources. Thus, a set of resources may be a set of computational resources. When a set of resources (or node) includes quantum computational hardware, the set of resources (or node) may be referred to as a quantum node (or quantum set of resources). Thus, the first set of resources 210 may be referred to as a first quantum node, the second set of resources 212 may be referred to as a second quantum node, the third set of resources 214 may be referred to as a third quantum node, and the fourth set of resources 216 may be a 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.

[0040] As shown via the coordinate system 208, each set of resources may be spatially separated from each of the other set of resources. Furthermore, the set of resources may be communicatively coupled by classical and / or quantum communication networks or channels, as indicated by the lines connecting (over both time and space) the various operations of the distributed computing environment 200. That is, the lines connecting the various operations may indicate communications (e.g., classical and / or quantum communications) between the sets of the set of nodes (or nodes). The distributed computing environment 200 includes a set of inputs 202, a set of computations 204, and a set of outputs 206. As shown in FIG.2A, the set of inputs 202, the set of computations 204, and the set of outputs 206 are distributed in time and space. Also, the set of inputs 202, the set of computations 204, and the set of outputs 206, as well as the communications between the sets of resources may be classical in nature, quantum in nature, or a combination thereof. The circular nodes of the set of computations may be functions (e.g., parameterized functions) and / or hardware resources that are allocated for computing the functions. The rectangular nodes of the set of inputs 202 and the set of outputs represent data (e.g., classicaland / or quantum data). When a computation of the set of computations 204 includes the processing of quantum information, the computation may be a quantum computation.

[0041] The following discussion focuses on compositional distributed learning (e.g., pipelining). Although the embodiments are not limited to compositional distributed learning, and other embodiments may be generalized to other forms of distributed learning. For instance, there are a number of strategies for distributing machine learning workloads that are influenced by the requirements of different applications and hardware constraints. FIG.2A shows a non-limiting example of splitting up a computational graph in a compositional fashion. Distributed, parameterized quantum circuits may be discussed in the context of the distributed computing environment 200. Such distributed quantum circuits can be used to perform inference over data when distributed in this way and can be trained using gradient methods. The embodiments can also be used to optimize models that use certain forms of data parallelism. In principle, such circuits could be implemented on quantum computers that are able to communicate quantum states.

[0042] Even for simple distributed quantum circuits, there is an exponential quantum advantage in communication for the problem of estimating the loss and the gradients of the loss with respect to the parameters. This additionally implies a privacy advantage. The embodiments employ a class of models that can efficiently approximate certain graph neural networks. The models of the embodiments maintain the exponential communication advantage, as well as achieve performance better (or at least comparable) to standard classical models on common node and graph classification benchmarks. For some distributed quantum circuits of the embodiments, there is an exponential advantage in communication for the entire training process, and not just for a single round of gradient estimation. This includes circuits for fine-tuning using pre-trained features. Some embodiments interleave multiple unitaries encoding nonlinear features of data. The ability to interleave multiple unitaries enables expressivity to grow exponentially with depth, and universal function approximation in some settings. Thus, the models of the embodiments are highly expressive in contrast to popular belief about linear restrictions in quantum neural networks.

[0043] Some embodiments employ a pipelining method for distributing a machine learning workload, in which different layers of a deep model are allocated to distinct hardware resources. Training and inference then require communication of features between nodes (or sets of resources). Pipelining enables flexible changes to the model architecture in a task-dependent manner, since subsets of a large model can be combined in an adaptive fashion to solve many downstream tasks. Additionally, pipelining allows sparse activation of a subset of a model required to solve a task and facilitates better use of heterogeneous compute resources since it does not require storing identical copies of a large model. Via pipelining, large (and distributed) modelsmay be fine-tuned to solve multiple tasks. In contrast to pipelining, data parallelism involves storing multiple copies of the model on different nodes, training each on a subset of the data and exchanging information to synchronize parameter updates. In practice, different parallelization strategies are combined to exploit trade-offs between latency and throughput in a task-dependent fashion.

[0044] Communication complexity is the study of distributed computational problems using a cost model that focuses on the communication required between players rather than the time or computational complexity. The key object of study in this area is the tree induced by a communication protocol whose nodes enumerate all possible communication histories and whose leaves correspond to the outputs of the protocol. The product structure induced on the leaves of this tree as a function of the inputs allows one to bound the depth of the tree from below, which gives an unconditional lower bound on the communication complexity. The power of replacing classical bits of communication with qubits has been studied. For certain problems such as Hidden Matching and a variant of classification with deep linear models, an exponential quantum communication advantage holds, while for other canonical problems such as disjointness only a polynomial advantage is possible.

[0045] Exponential advantages have been shown for the problem of sampling from a distribution defined by the solution to a linear regression problem. There are many models of both quantum and classical communication, and the embodiments apply to these many models. For instance, some non-limiting embodiments apply to randomized classical communication complexity. In some studies of these embodiments, the nodes are allowed to exchange random bits (of information) independent of their problem inputs and are allowed to output an incorrect answer with some probability (bounded away from 0.5 for a problem with binary output). It is also worth noting that communication advantages of the type demonstrated here can be naturally related to space advantages in streaming algorithms that may be of interest even in settings that do not involve distributed training.

[0046] The embodiments may employ networked quantum computers. For some quantum network architectures of the embodiments, the existence of a long-lasting fault tolerant quantum memory as a quantum repeater, may be an enabling component that lifts low-rate shared entanglement to a fully functional quantum network. As such, it is well motivated to consider potential communication advantages alongside computational advantages when talking about the applications of fault tolerant quantum computers.

[0047] FIG.2B shows a schematic of the bandwidth requirements for both classical and quantum communication protocols, when computing a gradient, for various embodiments. Moreparticularly, FIG.2B shows a schematic for classical communications for computing a gradient 220 during a training epoch. FIG.2B also shows a schematic for quantum communications for computing a gradient 230 during a training epoch.indicates a gradient estimator implemented at each of the sets of resources, ^^^^^^^^indicates forward features calculated at each of the set of resources, and ^^^^^^^^indicates backwards features calculated at each of the set of resources. Thus, a forward pass and a backward pass in each of the classical communications for computing a gradient 220 and quantum communications for computing a gradient 230 are indicated by the temporal sequence of forward and backward features. In the classical communications for computing a gradient 220, ℒ indicates a classical loss function, while in quantum communications for computing a gradient 230, ^^^^0indicates a quantum operator whose expectation value with respect to a quantum model gives the analogous loss function in the quantum case.

[0048] The size of arrows between the nodes for the calculations of the forward and backwards features indicates a relative “size difference” in the amount of information that must be communicated between the nodes. For instance, the larger arrow indicating the required bandwidth for classical information 222 in the classical communications for computing a gradient 220 is much larger than the smaller arrow indicating the required bandwidth for quantum information 232. In some embodiments, the bandwidth requirements for quantum information are exponentially smaller than the bandwidth requirements for classical information. Formalism for Distributed Learning with Quantum Resources

[0049] FIG.3 shows a schematic diagram for a distributed quantum circuit 300, according to various embodiments. For the purposes of simplicity, in distributed quantum circuit 300, only two nodes of a distributed computing environment are contemplated. However, in the general case, distributed quantum circuit 300 can include many more nodes. Also, in FIG.3 (and throughout), the canonical communications nomenclature for two parties in communication is adopted, and the two nodes are referred to as Alice and Bob (e.g., two spatially distributed parties exchanging (quantum) information). Thus, throughout, a set of resources (or nodes) may be referred to as “players” or “parties in communication.” A coordinate system 308 is provided for the distributed quantum circuit 300. Note that the axes in coordinate system 308 have been swapped, in comparison to coordinate system 208 of FIGS.2A-B. Throughout, ^^^^ is an unbounded positive integer that indicates the number of rounds of communication between the nodes (orplayers). For the non-limiting example of the distributed quantum circuit 300, ^^^^ = 2. Althoughthe embodiments are not so limited and embodiments where ^^^^ ≫ 2 are also contemplated.

[0050] In FIG.3, a communication problem with 2 nodes (e.g., players or parties), Alice and Bob, is considered. More specifically, Alice and Bob are working to compute a joint function determined through L rounds of communications with a normalized length N data input x ∈ SN−1, and N × N unitary matrices specifying the model given by Aℓ,Bℓ∈ U(N). The data and model matrices are distributed between Alice and Bob as Alice : x,{Aℓ} (Eq.1a) Bob : {Bℓ}, (Eq.1b)

[0051] Without loss of generality, it is assumed Aℓ, Bℓare each parameterized by Pℓparameters, and P ≡ maxℓ Pℓ. Denoting these by, for simplicity, some embodimentsassume that is also unitary, and similarly for Bℓ. While seemingly stringent, this is satisfied bymany of the most common quantum neural network architectures for example if Aℓ =and ^^^^^^^^^^^^^^^^are both unitary and Hermitian (e.g. Pauli matrices). Such models, or parameterized quantum circuits, are naturally amenable to implementation on quantum devices. The embodiments consider a general class of models of the form

[0052] Some embodiments estimate the quantity (or gradients of the following quantity) ^φ|P0|φ^ ≡ L, (Eq.3)

[0053] Where P0is a Pauli matrix that acts on the first qubit, after L rounds of communication, as depicted in FIG.3. Denote by ∇A(B)^φ|Z0|φ^, the entries of the gradient vector that correspond to {Aℓ}({Bℓ}).

[0054] At a high level, the state of Eq.3 is a model of multiple program multiple data (MPMD) or pipelined computation, in which different stages in a computation that has compositional structure are allocated distinct hardware resources. This is in contrast to parallelization by splitting up data across multiple identical copies of a model that exchange gradient information, known as same program multiple data (SPMD). Due to the relative inflexibility of SPMD, the latest generation of large models built to perform multiple heterogeneous tasks are designed based on MPMD. There are multiple advantages in using SPMD architectures. Such advantages include but are not limited to: (I) increased modularity, allowing different subsets of a model to be used for any given task without activating the full model. This is particularly useful for fine-tuning a model on many different tasks, and (II) the ability to efficiently utilize heterogeneous hardware.

[0055] Because of the cost associated with training large models, and the copious evidence that features learned to solve certain problems can be useful for solving many others, such MPMD architectures that enable greater modularity and flexibility are likely to become increasingly prominent. It is noted that the approach discussed below appears at least competitive with SPMD computation since this requires communicating the gradients themselves between players.

[0056] When training on batches of data, N would be analogous to the product of the batch size and input dimension. The interesting parameter regime for classical data and models is one where N,P are large, while L may only be moderate. For general unitaries, Pℓ= O(N2), which will also be the scaling of the number of parameters in fully connected networks. If {Aℓ} represent encodings of some input data then L is naturally related (typically exponentially) to the depth of the neural network with polynomial activations. In the case of transformers for example this depth is typically small. Across simple architectures that are amenable to analysis of expressive power, it is known to grow exponentially with network depth which helps explain the prevalence of this regime in practice. Some embodiments employ the expressive power of quantum circuits with similar structure. Such embodiments may employ data-dependent unitaries that act on a single qubit.

[0057] While the form of the model in Eq.2 may appear to be entirely linea0r, it can in fact, represent many useful classes of classical neural networks. In particular, {Aℓ} can be complex, nonlinear parameterized functions of the data. Some examples of which are provided below.

[0058] (i.) Feed-forward networks with polynomial activations: It is worth noting that networks with polynomial nonlinearities are universal function approximators, since roughly speaking depth can be used as a resource to compensate for the gap in expressivity between polynomial and non-polynomial functions.

[0059] (ii.) Unitary transformers: Transformers based on self-attention form the backbone of large language models and foundation models more generally. A self-attention layer, which is the central component of transformers, is a map between sequences in RS×N(where S is the sequence length) defined in terms of weight matrices WQ,WK,WV∈ RN×N, given by X′(X) = softmax,(Eq. 4)where softmax(x)i =a vector x, and acts row-wise on matrices. The softmax-basedattention matrix A(X) may be replaced with matrices that can be computed more efficiently, which can markedly improve the time complexity of inference and training without a significant effect on performance.

[0060] Considering a simplified model that does not contain the softmax operation and dropping normalization factors, the linear attention map is given by

[0061] Iterating this map K times (with different weight matrices at each layer) gives a function of the form: ^^^^(^^^^)lin (^^^^) = ^^^^^^^^0∏(3^^^^−1) / 2^^^^ℓ=1 ^^^^ ^^^^^^^^ℓ, (Eq.6) where the {Rℓ} matrices depend only on the trainable parameters. If these matrices are now constrained to be parameterized unitary matrices, and additionally replace XTX with a unitary matrix UXencoding features of the input sequence itself, then the i-th row of the output of this model is encoded in the amplitudes of a state of the form Eq.1, with L = (3K− 1) / 2 + 1, Bℓ= Zℓ, x = Xi, Aℓ= UX. This formulation is chosen primarily with simplicity of exposition in mind, and it may be unappealing that there is no explicit dependence on the sequence length S (and hence no direct analog of the attention matrix). However, it is straightforward to consider alternative formulations that do make this explicit. In such cases only a logarithmic dependence of the communication complexity on S is to be expected.

[0062] (iii.) Fine-tuning a pre-trained model: The simplest example of this is obtained by setting L = 1. For input x, the pre-trained model features are given by B1x, while the unitary A1represents a fine-tuning layer that takes the pre-trained features as input. B1can either be held constant while A1is updated by gradient descent, or both layers can be updated in tandem.

[0063] There is some evidence that certain types of features learned from data are localized in specific layers of deep models. For example, factual associations appear to be stored in the middle fully-connected layers of transformers. Thus, some embodiments employ more fine-grained schemes, where different players possess parameterized layers encoding different types of useful information.

[0064] Some embodiments employ expressive models of the form

[0065] Such models may be given the unitarity constraint of quantum mechanics on the matrices Aℓ(x) and Bℓ. Giving such a unitary constraint can depend on the encoding of the data that is chosen and the method of readout. For example, one could map bits to qubits 1-to-1 and encode any reversible classical function of data within the unitary matrix A(x) with the use of extra space qubits.

[0066] However, this may negate the possibility of any space advantages and does not provide any real computational advantage without additional processing. As noted above, other embodiments employ more generic functions in the amplitude and phase space, allowing for an exponential compression of the data into a quantum state. In such settings, it has been shown that Aℓ(x) can be made to express nearly arbitrary functions of the data x, paying costs that depend on the type of function. Some embodiments take a different approach to this question, inspired by work on the expressivity of certain classes of neural networks that is based on quantifying the strength of correlations between parts of the input that a network can efficiently represent. One example metric, separation rank (denoted sep), may be employed. For a function f of two variables y and z, it is defined by:

[0067] If, for example, f cannot represent any correlations between y and z, then sep(f) = 1. When computed for certain classes of neural networks, y and z are taken to be subsets of a high- dimensional input. The separation rank can be used for example to quantify the inductive bias of convolutional networks towards learning local correlations, the effect of depth in recurrent networks, and the ability of transformers to capture correlations across sequences as a function of their depth and width.

[0068] For a linear transformer written in the form Eq.6, it is known that the separation rank depends exponentially on the depth of the compositional circuit for almost all values of the inputs and parameters, until some threshold set by the network width. In the quantum setting it is therefore natural to conjecture that, analogously, log(sep(φ;I,J)) = Θ(L). This appears trivial for a general Aℓwhich would require specializing. One drawback with this type of result is that it is not robust and so would need to be strengthened to deal with measurement error or finite precision.

[0069] Consider the expressivity of circuits with multiple data-dependent unitaries interleaved with parameterized unitaries via analysis of the Fourier spectrum of such models. An asymptotic universality result may be achieved for shallow models of this form that rely on using expressive data encodings. The explicit class of Hamiltonians they consider however has the number of qubits scaling exponentially with the depth of the model. It is possible to obtain a more general result. (Note also the connection to QSP, where the parameterized matrices are single qubit rotations and expressivity only grows linearly with depth).

[0070] Given two scalar variables x and y, definefor ℓ ∈ {1, ... ,L}. We also consider a general class of parameterized unitaries {Bℓ} that are independent of the “features” {λℓi} and “inputs” x and y. Denoted by |φ(x)^, |φ(x,y)^, the states defined by interleaving these unitaries in the manner of Eq.2, and by L1, L2the corresponding loss functions when measuring X0, we have the following expressivity result.

[0071] Lemma 1. For {λℓi} drawn i.i.d. from any continuous distribution and parameterized unitaries {Bℓ} such that the real and imaginary parts of each entry in these matrices is a real analytic function of parameters Θ drawn from a subset of RPL, aside from a set of measure10)

[0072] This almost saturates the upper bound on the number of frequencies that can be expressed by a circuit of various forms. This result can be trivially generalized to the setting where y and z are replaced by sequences of scalars / vectors. The separation rank implies that complex correlations between different parts of the sequence can in principle be represented by such circuits. The constraint on {Bℓ} is quite mild and applies to standard choices of parameterized unitaries.

[0073] The main shortcoming of a result such as Lemma 1 is that it is not robust to measurement error as it is based on constructing states that are equal weight superpositions of an exponential number of terms. It is best viewed as a necessary but insufficient condition for guaranteeing that this model class is highly expressive. It is also natural to ask whether it is possible to easily express any single combination of features from different layers in a manner that is robust to measurement error. It can be shown that it is indeed easy to express O(NL) different frequencies in this way.

[0074] An interesting question that may require a more sophisticated approach is whether these types of circuits can serve as efficient universal function approximators. Since an L layer circuit with {Aℓ} given by Eq.9 will have at most O(N2L) independent degrees of freedom in the trainable unitaries {Bℓ}, it will only be possible to control the O(N2L−1) Fourier coefficients implied by Lemma 1 independently if L = O(log N). Similar scaling has been observed for classical modelssuch as linear transformers. It may be useful to consider other feature maps in this context, including parameterized ones.

[0075] Given the inputs of Eq.1, the embodiments contemplate two problems: Problem 1 (Distributed Inference). Alice and Bob each compute an estimate of ^φ|Z0|φ^ with additive error ε. Problem 2 (Distributed Gradient Estimation). Alice computes an estimate of ∇A^φ|Z0|φ^, while Bob computes an estimate of ∇B^φ|Z0|φ^, with additive error ε in L∞.

[0076] Lemma 2. Problem 1 can be solved by communicating O(log N) qubits over O(L / ε2) rounds.

[0077] Problem 2 can be solved with probability greater than 1 − δ by communicating log(^^^^ / ^^^^) / ^^^^4) qubitsThe time and space complexity of the algorithm is√^^^^^^^^poly(^^^^, log^^^^, ^^^^−1, log(1 / ^^^^)).

[0078] Using immediate reductions from known problems in communication complexity, we can show that lots of classical communication is required to solve the above problems. One may additionally wonder whether the quantum algorithm is tight in terms of the total number of qubits communicated and the number of rounds. It is shown that at least Ω(L) rounds may be necessary to obtain an exponential communication advantage.

[0079] Lemma 3. The classical communication complexity of Problem 1 and Problem 2 isΩ (max (√^^^^, ^^^^). Any quantum(classical) protocol for Problem 1 or Problem 2 requires eitherΩ(L) rounds of communication or Ω(N / L4) qubits(bits) of communication.

[0080] Note that the lower bound requires the ability to apply a general orthogonal matrix, which would imply a circuit of depth poly(N). This unitary need not be parameterized or trainable though. One can ask whether the lower bound can be strengthened to apply to more restricted classes of unitaries as well. Combining these two results, in the regime where L = O(polylog(N)), one obtains an exponential advantage in communication complexity at the cost of a polynomial overhead in computational complexity.

[0081] Quantum advantage is possible in these problems because there is a bound on the complexity of the final output, whether it be correlated elements of the gradient up to some finite error or the low-dimensional output of a model. This might lead one to believe that whenever the output takes such a form, encoding the data in the amplitudes of a quantum state will trivially give an exponential advantage in communication complexity. We show however that the situation is slightly more nuanced, by considering the problem of inference with a linear model.

[0082] Lemma 4. For the problem of distributed linear classification, any quantum advantage in communication is at most polynomial.

[0083] The problem of estimating a single gradient vector has been discussed above. It is natural to also consider when these or other gradient estimators can be used to efficiently solve an optimization problem. Another algorithm that requires a single quantum measurement per iteration is presented in FIG.4A. FIG.4A shows a non-limiting example of a first algorithm 400 that is a quantum tomography algorithm, according to various embodiments. More particularly, the first algorithm 400 is directed towards shadow tomographic distributed gradient descent. The second intermediate output is determined at the third quantum node. Determining the second intermediate result is based on the encoding of the second intermediate output and the quantum tomography algorithm. A result of the distributed computation is computed at the third quantum node. Calculating the result of the distributed computation is based on a combination of the first intermediate output and the second intermediate output. The distributed computation includes providing the first input and the second input to the model.

[0084] The below algorithm may solve certain convex problems efficiently, as well as an application of shadow tomography to fine-tuning.

[0085] The following is a generalization of algorithms in a distributed setting. Consider the case where Aℓare product of rotations for all ℓ, namelywhere ^^^^ℓ^^^^^^^^are Pauli matrices acting on all qubits, and similarly for Bℓ. These can also be interspersed with other non-trainable unitaries. Denoted by β an 2PL-dimensional vector withelements . The quantity ∥β∥1 is the total evolution time if we interpret the state |φ^ as a sequenceof Hamiltonians applied to the initial state |x^.

[0086] FIG.4B shows a non-limiting second algorithm 420 that is directed towards distributed probabilistic coordinate descent, according to various embodiments. Given distributed states of the above form, optimization over Θ can be performed using the second algorithm 420, or similar algorithms. A Hermitian measurement is defined aswith eigenvalues in {−1,1}. Note that . Now consider a gradient estimatorthat first samples (Q, ℓ, i) with probabilitythen returns a one-sparse vector ℊ^^^^ = ^^^^^^^^^^^^^^^^�^^^^^^^^�‖^^^^‖ ^^^^, where m is the result of a single measu� ^^^^ℓ^^^^ ℓ^^^^ 1 rement of ^^^^ℓ^^^^using the state. For this estimator:where the expectation is taken over both the index sampling process and the quantum measurement. The procedure generates a valid gradient estimator.

[0087] In order to show convergence, one simply notes that by construction, ∥g∥2= ∥β∥1. Itcan be shown that, with an appropriately chosen step size, Algorithm 2 achieves ^^^^ℒ(^^^^) −ℒ(^^^^∗) ≤ ^^^^0 for a convex ℒ usingqueries. For a λ-strongly convex ℒ, only (Eq.15) queries are required. The pre-processing in step 1 of Algorithm 2 requires time O(P log P) and subsequently enables sampling in time O(1) using e.g., Walker (1974) (even a simpler strategy like sorting and inverse CDF sampling will enable sampling with cost O(log P)).

[0088] This implies an exponential advantage in communication for the entire training process as long aspolylog(N). Such circuits either have a small number oftrainable parameters (P =O(polylog(N))), or else depend weakly on eachparameter (e.g., for arbitrary P). Nevertheless, over a convex region the loss can rapidly change by an O(1) amount. One may also be concerned that in the setting= polylog(N) only a logarithmic number of parameters is updated during the entire training process and so the total effect of the training process may be negligible. It is important to note however that each such sparse update depends on the structure of the entire gradient vector as seen in the sampling step. In this sense the algorithm is reminiscent of a probabilistic version of coordinatedescent, since the probability of updating a coordinate is proportional to the magnitude ofthe corresponding element in the gradient (actually serving as an upper bound for it).

[0089] The time complexity of a single iteration of this algorithm is proportional to a forward pass, and so matches the scaling of classical backpropagation. This is in contrast to the polynomial overhead of shadow tomography (e.g., see Theorem 1). It only requires a single measurement per iteration and minimal additional overhead, suggesting that it might be viable to implement on noisy near-term devices.

[0090] Consider a model of depth L where only ALis parameterized and denote this model by |φf^. The circuit up to that unitary could include multiple data-dependent unitaries and so could represent complex features in the data. Training only the final layer in this manner is one method used by various embodiments for fine-tuning a pre-trained model. In this setting,is the expectation value of ELiA using the state

[0091] Since the observables in the shadow tomography problem can be chosen in an online fashion, and adaptively based on previous measurements, we can simply define a stream of measurement operators by measuring P observables to estimate the gradients w.r.t. an initial set of parameters, updating these parameters using gradient descent with step size η, and defining a new set of observables using the updated parameters. Repeating this for T iterations gives a total of PT observables. By the scaling in Lemma 2, the total communication needed is ^^�^^(log N log4(TP)) log(L / δ) / ε4) over O(L) rounds (since only O(L) rounds are needed to create copies of .) This implies an exponential advantage in communication for the entire training process (under the reasonable assumption T = O(poly(N,P))), despite the additional stochasticity introduced by theneed to perform quantum measurements. For example, assume one has a boundIf thecircuit is comprised of unitaries with Hermitian derivatives, this holds with K = PL. In that case, denoting by g the gradient estimator obtained by shadow tomography,

[0092] It can be shown that for an appropriately chosen step size, if L is convex one canfind parameter values ^^�^^ such thatℒ�(�^�^^�^�) − ℒ(^^^^∗) ≤ ^^^^0 (Eq. 19)

[0093] Using(Eq.20) iterations of gradient descent. Similarly, if L is λ-strongly convex theniterations are sufficient. In both cases an exponential advantage is achieved for the optimization process as a whole.

[0094] The results of the above discussion relied on so called sublinear convergence rates for general stochastic optimization of convex functions. Using additional structure, stochastic gradients can be used to obtain linear convergence (meaning the error scales like rTfor some r < 1). This is possible for example when subsampling is the source of stochasticity, or with occasional access to noiseless gradients to implement a variance reduction scheme, neither of which seem applicable to the setting at hand. It is an interesting open question to ascertain whether there is a way to exploit the structure of quantum circuits to obtain linear convergence rates using novel algorithms. Aside from advantages in time complexity, this could imply an exponential advantage in communication for a more general class of circuits.

[0095] Conversely, it is also known that given only black-box access to a noisy gradient oracle, an information-theoretic lower bound of Ω(1 / T) on the error holds given T oracle queries, precluding linear convergence without additional structure, even for strongly convex objectives. These results may be used to show optimality of algorithms that rely on the standard variational circuit optimization paradigm that involves making quantum measurements at every iteration and using these to update the parameters. This implies that linear convergence is possible if the entire optimization process is performed coherently.

[0096] In addition to an advantage in communication complexity, the quantum algorithm outlined above has an inherent advantage in terms of privacy. 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. It follows immediately that since the above algorithm requires exchanging a logarithmic number of copies of states over O(log N) qubits, even if all the communication between the two players is intercepted, an attacker cannot extract more than a logarithmic number of bits of classical information about the input data or model parameters. Specifically,

[0097] Corollary 1. If all the communication between Alice and Bob is intercepted by an attacker, the attacker cannot extract more than O˜(L2log2N log4P log(L / δ) / ε4) bits of classical information.

[0098] It is also interesting to ask how much information either Bob or Alice can extract about the inputs of the other player by running the protocol. If this amount is logarithmic as well, it provides additional privacy to both the model owner and the data owner. It allows two actors who do not necessarily trust each other, or the channel through which they communicate, to cooperate in jointly training a distributed model or using one for inference while only exposing a vanishing fraction of the information they hold.

[0099] It is also worth mentioning that data privacy is also guaranteed in a scenario where the user holding the data also specifies the processing done on the data. In this setting, Alice holds both data x and a full description of the unitaries she wishes to apply to her state. She can send Bob a classical description of these unitaries, and so long as the data and features are communicated in the form of quantum states, only a logarithmic amount of information can be extracted about them. In this setting there is of course no advantage in communication complexity, since the classical description of the unitary will scale like poly(N, P). Methods

[0100] FIG.5 depicts a flow diagram of an example method 500 for performing a distributed computation via a model and a distributed quantum computation system (DQCS), according to various embodiments. The DQCS comprises of a set of quantum nodes and a quantum communication network that communicatively couples quantum nodes of the set of quantum nodes. At block 502, a first quantum node of the set of quantum nodes receives a first input to the model. At block 504, the first quantum node performs a first quantum computation based on the first input and a first component of the model that is implemented by the first node. Performing the first quantum computation includes generating an encoding of a first intermediate output of the model. At block 506, a second quantum node of the set of nodes receives a second input to the model. At block 508, the second quantum node performs a second quantum computation based on the second input and a second component of the model that is implemented by the second quantum node. Performing the second quantum computation includes generating an encoding of a second intermediate output of the model. At block 510, the encoding of the first intermediate output is transmitted from the first quantum node to a third quantum node of the set of quantum nodes, via the quantum communication network. At 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. At 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 a quantum tomography algorithm. Atblock 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. At block 518, a result of the distributed computation is determined at the third quantum node. Determining the result of the distributed computation is based on a combination of the first intermediate output and the second intermediate output. The distributed computation includes providing the first input and the second input to the model.

[0101] In various embodiments, the model is a neural network implemented machine- learned model. The first input and the second input may be included in a training dataset for the model. The result of the distributed computation may include a gradient of a loss function that is evaluated based on the first input and the second input being provided to the model. The method may further include performing, at the third quantum node, a descent of the gradient of the loss function. A set of parameters of the model may be updated based on the descent of the gradient of the loss function. In some embodiments, updating the set of parameters of the model includes training the model. In other embodiments, updating the set of parameters of the model includes fine-tuning the model. The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0102] In some embodiments, performing the first quantum computation includes preparing a first quantum state of a first set of qubits of the first quantum node such that the first quantum state of the first set of qubits stores the encoding of the first intermediate output of the model. Performing the second quantum computation may include preparing a second quantum state of a second set of qubits of the second quantum node such that the second quantum state of the second set of qubits stores the encoding of the second intermediate output of the model.

[0103] In such embodiments, transmitting the encoding of the first intermediate output includes transmitting the first quantum state of the first set of qubits, via the quantum communication network, from the first quantum node to the third quantum node. Transmitting the encoding of the second intermediate output may include transmitting the second quantum state of the second set of qubits, via the quantum communication network, from the second quantum node to the third quantum node.

[0104] In these embodiments, transmitting the first quantum state of the first set of qubits includes transmitting a plurality of copies of the first quantum state, via the quantum communication network, from the first quantum node to the third quantum node. Transmitting the second quantum state of the first set of qubits may include transmitting a plurality of copies of the second quantum state, via the quantum communication network, from the second quantum node to the third quantum node. Determining the first intermediate output of the model may includeproviding the plurality of copies of the first quantum state to the quantum tomography algorithm. Determining the second intermediate output may include providing the plurality of copies of the second quantum state to the quantum tomography algorithm. The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0105] 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 quantum states. The model may be a trained machine-learned model. In such embodiments, the result of the distributed computation includes an inference-time prediction of the trained machine-learned model based on the first input.

[0106] Other embodiments are directed to a method that includes receiving a first input at a distributed quantum computing system (DQCS). The DQCS may have a plurality of quantum nodes and a quantum communication network. A first quantum computation is performed at a first quantum node of the plurality of nodes. The first quantum computation may be based on the first input and a model. An output of the first quantum computation may be communicated to a second quantum node of the plurality of quantum nodes over the quantum communication network. A result of a distributed computation is determined and / or computed at the second quantum node. Determining the result of the distributed computation may be based on the output of the first quantum computation and a quantum tomography algorithm implemented at the second quantum node. The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0107] Such a method may further include receiving a second input at the DQCS. A second quantum computation may be performed at a third quantum node of the plurality of nodes. Performing the second quantum computation may be based on the second input and the model. An output of the second quantum computation may be communicated to the second quantum node over the quantum communication network. The result of the distributed computation may be determined further based on the output of the second quantum computation and the quantum tomography algorithm.

[0108] Another method may be directed to performing a distributed computation via a model and a distributed quantum computation system (DQCS). The DQCS may comprise of a first quantum node, a second quantum node, and a first quantum communication channel that communicatively couples the first quantum node and the second quantum node. The method includes receiving, at the first quantum node, a first input. A first component of the model is implemented at the first quantum node. Via the first quantum communication channel, an encoding of the first input is transmitted from the first quantum node to the second quantum node. A first quantum computation is performed at the second quantum node. Performing the firstquantum computation is based on the encoding of the first input and a second component of the model that is implemented at the second quantum node. Performing the first quantum computation generates a first output. An encoding of the first output is transmitted from the second quantum node to the first quantum node. A second quantum computation is performed at the first quantum node. Performing the second quantum computation is based on the encoding of the first output and the first component of the model, wherein performing the second quantum computation generates a second output. Additional Embodiments

[0109] One non-limiting embodiment includes a method. The method includes receiving a first input at a distributed quantum computing system (DQCS). The DQCS has a plurality of quantum nodes and a quantum communication network. The method further includes performing a first quantum computation at a first quantum node of the plurality of nodes based on the first input and a model. The method further includes communicating an output of the first quantum computation to a second quantum node of the plurality of quantum nodes over the quantum communication network. The method further includes determining a result of a distributed computation based on the output of the first quantum computation and a quantum tomography algorithm implemented at the second quantum node.

[0110] The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0111] The method may further include receiving a second input at the DQCS. A second quantum computation is performed at a third quantum node of the plurality of nodes based on the second input and the model. An output of the second quantum computation is communicated to the second quantum node over the quantum communication network. Determining the result of the distributed computation is further based on the output of the second quantum computation and the quantum tomography algorithm.

[0112] The model may be a neural network that implements a machine-learned model. The first input and the second input may be included in a training dataset for the model. The result of the distributed computation may include a gradient of a loss function that is evaluated based on the first input and the second input being provided to the model.

[0113] The method may further include performing, at the second quantum node, a descent of the gradient of the loss function. A set of parameters of the model is updated based on the descent of the gradient of the loss function.

[0114] In some embodiments, updating the set of parameters of the model may include training the model.

[0115] In some embodiments, updating the set of parameters of the model includes fine- tuning the model.

[0116] Performing the first quantum computation may include preparing a first quantum state of a first set of qubits of the first quantum node such that the first quantum state of the first set of qubits stores the encoding of the output of the first quantum computation.

[0117] Communicating the output of the first quantum computation may include communicating the first quantum state of the first set of qubits, via the quantum communication network, from the first quantum node to the second quantum node.

[0118] The quantum communication network can transmit quantum states via at least one of a quantum teleportation algorithm or photons that encode quantum states.

[0119] Another method is a method for performing a distributed computation via a model and a distributed quantum computation system (DQCS). The DQCS comprises a set of quantum nodes and a quantum communication network that communicatively couples quantum nodes of the set of quantum nodes. The method includes receiving, at a first quantum node of the set of quantum nodes, a first input to the model. The method includes performing, at the first quantum node, a first quantum computation based on the first input and a first component of the model that is implemented by the first node, wherein performing the first quantum computation includes generating an encoding of a first intermediate output of the model. The method includes receiving, at a second quantum node of the set of nodes, a second input to the model. The method includes performing, at the second quantum node, a second quantum computation based on the second input and a second component of the model that is implemented by the second quantum node, wherein performing the second quantum computation includes generating an encoding of a second intermediate output of the model. The method includes transmitting, via the quantum communication network, the encoding of the first intermediate output from the first quantum node to a third quantum node of the set of quantum nodes. The method includes transmitting, via the quantum communication network, the encoding of the second intermediate output from the second quantum node to the third quantum node. The method includes determining, at the third quantum node, the first intermediate output based on the encoding of the first intermediate output and a quantum tomography algorithm. The method includes determining, at the third quantum node, the second intermediate output based on the encoding of the second intermediate output and the quantum tomography algorithm. The method includes calculating, at the third quantum node, a result of the distributed computation based on a combination of the first intermediate output andthe second intermediate output, wherein the distributed computation includes providing the first input and the second input to the model.

[0120] The model may be a neural network implemented machine-learned model, the first input and the second input are included in a training dataset for the model, and the result of the distributed computation includes a gradient of a loss function that is evaluated based on the first input and the second input being provided to the model.

[0121] The method may include performing, at the third quantum node, a descent of the gradient of the loss function. A set of parameters of the model is updated based on the descent of the gradient of the loss function.

[0122] Updating the set of parameters of the model may include training the model.

[0123] Updating the set of parameters of the model may include fine-tuning the model.

[0124] The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0125] Performing the first quantum computation may include preparing a first quantum state of a first set of qubits of the first quantum node such that the first quantum state of the first set of qubits stores the encoding of the first intermediate output of the model and performing the second quantum computation includes preparing a second quantum state of a second set of qubits of the second quantum node such that the second quantum state of the second set of qubits stores the encoding of the second intermediate output of the model.

[0126] Transmitting the encoding of the first intermediate output may include transmitting the first quantum state of the first set of qubits, via the quantum communication network, from the first quantum node to the third quantum node, and wherein transmitting the encoding of the second intermediate output includes transmitting the second quantum state of the second set of qubits, via the quantum communication network, from the second quantum node to the third quantum node.

[0127] Transmitting the first quantum state of the first set of qubits may include transmitting a plurality of copies of the first quantum state, via the quantum communication network, from the first quantum node to the third quantum node, and wherein transmitting the second quantum state of the first set of qubits includes transmitting a plurality of copies of the second quantum state, via the quantum communication network, from the second quantum node to the third quantum node.

[0128] Determining the first intermediate output of the model may include providing the plurality of copies of the first quantum state to the quantum tomography algorithm, and wherein determining the second intermediate output includes providing the plurality of copies of the second quantum state to the quantum tomography algorithm.

[0129] The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0130] In some embodiments, the quantum communication network may transmit quantum states via a quantum teleportation algorithm.

[0131] In some embodiments, the quantum communication network transmits quantum states via photons that encode quantum states.

[0132] The model may be a trained machine-learned model and the result of the distributed computation includes an inference-time prediction of the trained machine-learned model based on the first input.

[0133] Another method is for performing a distributed computation via a model and a distributed quantum computation system (DQCS) comprising of a first quantum node, a second quantum node, and a first quantum communication channel communicatively coupling the first quantum node and the second quantum node. The method includes receiving, at the first quantum node, a first input, wherein a first component of the model is implemented at the first quantum node. The method includes transmitting, via the first quantum communication channel, an encoding of the first input from the first quantum node to the second quantum node. The method includes performing, at the second quantum node, a first quantum computation based on the encoding of the first input and a second component of the model located at the second quantum node, wherein performing the first quantum computation generates a first output. The method includes transmitting, via the first quantum communication channel, an encoding of the first output from the second quantum node to the first quantum node. The method includes performing, at the first quantum node, a second quantum computation based on the encoding of the first output and the first component of the model, wherein performing the second quantum computation generates a second output.

[0134] The DQCS may further include a third quantum node and a second quantum communication channel that communicatively couples the second quantum node and the third quantum node. The method may further include receiving, at the first quantum node, a first input, wherein a first component of the model is implemented at the first quantum node. The method further includes transmitting, via the first quantum communication channel, an encoding of the first input from the first quantum node to the second quantum node. The method includes performing, at the second quantum node, a first quantum computation based on the encoding of the first input and a second component of the model located at the second quantum node, wherein performing the first quantum computation generates a first output. The method includes transmitting, via the first quantum communication channel, an encoding of the first output from the second quantum node tothe first quantum node. The method includes performing, at the first quantum node, a second quantum computation based on the encoding of the first output and the first component of the model, wherein performing the second quantum computation generates a second output.

[0135] One embodiment includes a distributed quantum computing system (DQCS). The DQCS includes a set of quantum nodes that includes a first quantum node and a second quantum node. The DQCS includes a quantum communication network that communicatively couples the quantum nodes of the set of quantum nodes. The DQCS includes one or more processors. The DCCS includes one or more memory devices, the one or more memory devices storing computer- readable instructions that when executed by the one or more processors cause the one or more processors to perform operations for performing a distributed computation via a model. The operations include receiving, at a first quantum node of the set of quantum nodes, a first input to the model. The operations include performing, at the first quantum node, a first quantum computation based on the first input and a first component of the model that is implemented by the first node, wherein performing the first quantum computation includes generating an encoding of a first intermediate output of the model. The operations include receiving, at a second quantum node of the set of nodes, a second input to the model. The operations include performing, at the second quantum node, a second quantum computation based on the second input and a second component of the model that is implemented by the second quantum node, wherein performing the second quantum computation includes generating an encoding of a second intermediate output of the model. The operations include transmitting, via the quantum communication network, the encoding of the first intermediate output from the first quantum node to a third quantum node of the set of quantum nodes. The operations include transmitting, via the quantum communication network, the encoding of the second intermediate output from the second quantum node to the third quantum node. The operations include determining, at the third quantum node, the first intermediate output based on the encoding of the first intermediate output and a quantum tomography algorithm. The operations include determining, at the third quantum node, the second intermediate output based on the encoding of the second intermediate output and the quantum tomography algorithm. The operations include calculating, at the third quantum node, a result of the distributed computation 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.

[0136] The quantum tomography algorithm may be a shadow quantum tomography algorithm.

[0137] One method includes one or more aspects of any of the embodiments provided herein, or variants thereof.

[0138] Another embodiment includes a distributed quantum computing system (DQCS) comprising one or more aspects of any of the embodiments provided herein, or variants thereof.

[0139] One example aspect of the present disclosure is directed to a method for performing a distributed computation. The distributed computation is performed via a model and a distributed quantum computation system (DQCS). The DQCS comprising a set of quantum nodes and a quantum communication network that communicatively couples quantum nodes of the set of quantum nodes. The method includes receiving, at a first quantum node of the set of quantum nodes, a first input to the model. A first quantum computation is performed at the first quantum node. Performing the first quantum computation is based on the first input and a first component of the model that is implemented by the first node. Performing the first quantum computation includes generating an encoding 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. A second quantum computation is performed at the second quantum node. Performing the second quantum computation is based on the second input and a second component of the model that is implemented by the second quantum node. Performing the second quantum computation includes generating an encoding of a second intermediate output of the model. The encoding of the first intermediate output is transmitted via the quantum communication network from the first quantum node to a third quantum node of the set of quantum nodes. The encoding of the second intermediate output is transmitted via the quantum communication network from the second quantum node to the third quantum node. 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 a quantum tomography algorithm.

[0140] One example aspect of the present disclosure is directed to a method for performing a distributed computation. The distributed computation is performed via a model and a distributed quantum computation system (DQCS). The DQCS comprising a set of quantum nodes and a quantum communication network that communicatively couples quantum nodes of the set of quantum nodes. The method includes receiving, at a first quantum node of the set of quantum nodes, a first input to the model. A first quantum computation is performed at the first quantum node. Performing the first quantum computation is based on the first input and a first component of the model that is implemented by the first node. Performing the first quantum computation includes generating an encoding 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. A second quantumcomputation is performed at the second quantum node. Performing the second quantum computation is based on the second input and a second component of the model that is implemented by the second quantum node. Performing the second quantum computation includes generating an encoding of a second intermediate output of the model. The encoding of the first intermediate output is transmitted via the quantum communication network from the first quantum node to a third quantum node of the set of quantum nodes. The encoding of the second intermediate output is transmitted via the quantum communication network from the second quantum node to the third quantum node. 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 a quantum tomography algorithm. The second intermediate output is determined at the third quantum node. Determining the second intermediate result is based on the encoding of the second intermediate output and the quantum tomography algorithm. A result of the distributed computation is computed at the third quantum node. Calculating the result of the distributed computation is based on a combination of the first intermediate output and the second intermediate output. The distributed computation includes providing the first input and the second input to the model.

[0141] Implementations of the digital, classical, and / or quantum subject matter and the digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-implemented digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The term “quantum computing systems” may include, but is not limited to, quantum computers / computing systems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0142] Implementations of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, data processing apparatus. The digital and / or quantum computer storage medium can be a machine- readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits / qubit structures, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially-generated propagated signal that is capable of encoding digital and / or quantum information (e.g., a machine-generatedelectrical, optical, or electromagnetic signal) that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus.

[0143] The terms quantum information and quantum data refer to information or data that is carried by, held, or stored in quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states (e.g., qudits) are possible.

[0144] The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, or multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, 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 apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0145] A digital or classical computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, ordeclarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL, Quipper, Cirq, etc..

[0146] A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, sub-programs, or portions of code. A digital and / or quantum computer program can be deployed to be executed on one digital or one quantum computer or on multiple digital and / or quantum computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data.

[0147] The processes and logic flows described in this specification can be performed by one or more programmable digital and / or quantum computers, operating with one or more digital and / or quantum processors, as appropriate, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers.

[0148] For a system of one or more digital and / or quantum computers or processors to be “configured to” or “operable to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more digital and / or quantum computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by digital and / or quantum data processing apparatus, cause the apparatus to perform the operations or actions. A quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions.

[0149] Digital and / or quantum computers suitable for the execution of a digital and / or quantum computer program can be based on general or special purpose digital and / or quantum microprocessors or both, or any other kind of central digital and / or quantum processing unit. Generally, a central digital and / or quantum processing unit will receive instructions and digitaland / or quantum data from a read-only memory, or a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof.

[0150] Some example elements of a digital and / or quantum computer are a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a digital and / or quantum computer will also include, or be operatively coupled to receive digital and / or quantum data from or transfer digital and / or quantum data to, or both, one or more mass storage devices for storing digital and / or quantum data, e.g., magnetic, magneto-optical disks, or optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices.

[0151] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non- volatile digital and / or quantum memory, media and memory devices, including by way of example 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; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence.

[0152] Control of the various systems described in this specification, or portions of them, can be implemented in a digital and / or quantum computer program product that includes instructions that are stored on one or more tangible, non-transitory machine-readable storage media, and that are executable on one or more digital and / or quantum processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or electronic system that may include one or more digital and / or quantum processing devices and memory to store executable instructions to perform the operations described in this specification.

[0153] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementationsseparately or in any suitable sub combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0154] Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations described above should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0155] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

Claims

WHAT IS CLAIMED IS:

1. 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; performing a first quantum computation at a first quantum node of the plurality of quantum nodes based on the first input and a model; communicating an output of the first quantum computation to a second quantum node of the plurality of quantum nodes over the quantum communication network; and determining a result of a distributed computation based on the output of the first quantum computation and a quantum tomography algorithm implemented at the second quantum node.

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

3. The method of claim 1, further comprising: receiving a second input at the DQCS; performing a second quantum computation at a third quantum node of the plurality of quantum nodes based on the second input and the model; communicating an output of the second quantum computation to the second quantum node over the quantum communication network; and determining the result of the distributed computation further based on the output of the second quantum computation and the quantum tomography algorithm.

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

5. The method of claim 4, further comprising: performing, at the second quantum node, a descent of the gradient of the loss function; andupdating a set of parameters of the model based on the descent of the gradient of the loss function.

6. The method of claim 5, wherein updating the set of parameters of the model includes training the model.

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

8. The method of claim 1, wherein performing the first quantum computation includes preparing a first quantum state of a first set of qubits of the first quantum node such that the first quantum state of the first set of qubits stores an encoding of the output of the first quantum computation.

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

10. The method of claim 1, wherein the quantum communication network transmits quantum states via at least one of a quantum teleportation algorithm or photons that encode quantum states.

11. A method for performing a distributed computation via a model and a distributed quantum computation system (DQCS) comprising a set of quantum nodes and a quantum communication network that communicatively couples quantum nodes of the set of quantum nodes, the method comprising: receiving, at a first quantum node of the set of quantum nodes, a first input to the model; performing, at the first quantum node, a first quantum computation based on the first input and a first component of the model that is implemented by the first quantum node, wherein performing the first quantum computation includes generating an encoding of a first intermediate output of the model; receiving, at a second quantum node of the set of nodes, a second input to the model; performing, at the second quantum node, a second quantum computation based on the second input and a second component of the model that is implemented by the second quantumnode, wherein performing the second quantum computation includes generating an encoding of a second intermediate output of the model; transmitting, via the quantum communication network, the encoding of the first intermediate output from the first quantum node to a third quantum node of the set of quantum nodes; transmitting, via the quantum communication network, the encoding of the second intermediate output from the second quantum node to the third quantum node; determining, at the third quantum node, the first intermediate output based on the encoding of the first intermediate output and a quantum tomography algorithm; determining, at the third quantum node, the second intermediate output based on the encoding of the second intermediate output and the quantum tomography algorithm; and calculating, at the third quantum node, a result of the distributed computation 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 of claim 11, wherein the model is a neural network implemented machine-learned model, the first input and the second input are included in a training dataset for the model, and the result of the distributed computation includes a gradient of a loss function that is evaluated based on the first input and the second input being provided to the model.

13. The method of claim 12, further comprising: performing, at the third quantum node, a descent of the gradient of the loss function; and updating a set of parameters of the model based on the descent of the gradient of the loss function.

14. The method of claim 13, wherein updating the set of parameters of the model includes training the model.

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

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

17. The method of claim 11, wherein performing the first quantum computation includes preparing a first quantum state of a first set of qubits of the first quantum node such that the first quantum state of the first set of qubits stores the encoding of the first intermediate output of the model and performing the second quantum computation includes preparing a second quantum state of a second set of qubits of the second quantum node such that the second quantum state of the second set of qubits stores the encoding of the second intermediate output of the model.

18. The method of claim 17, wherein transmitting the encoding of the first intermediate output includes transmitting the first quantum state of the first set of qubits, via the quantum communication network, from the first quantum node to the third quantum node, and wherein transmitting the encoding of the second intermediate output includes transmitting the second quantum state of the second set of qubits, via the quantum communication network, from the second quantum node to the third quantum node.

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

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

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

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

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

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

25. A method for performing a distributed computation via a model and a distributed quantum computation system (DQCS) comprising of a first quantum node, a second quantum node, and a first quantum communication channel communicatively coupling the first quantum node and the second quantum node, the method comprising: receiving, at the first quantum node, a first input, wherein a first component of the model is implemented at the first quantum node; transmitting, via the first quantum communication channel, an encoding of the first input from the first quantum node to the second quantum node; performing, at the second quantum node, a first quantum computation based on the encoding of the first input and a second component of the model located at the second quantum node, wherein performing the first quantum computation generates a first output; transmitting, via the first quantum communication channel, an encoding of the first output from the second quantum node to the first quantum node; and performing, at the first quantum node, a second quantum computation based on the encoding of the first output and the first component of the model, wherein performing the second quantum computation generates a second output.

26. The method of claim 25, wherein the DQCS further comprises a third quantum node and a second quantum communication channel that communicatively couples the second quantum node and the third quantum node, the method further comprising: receiving, at the third quantum node, a second input, wherein a third component of the model is located at the third quantum node; transmitting, via the second quantum communication channel, an encoding of the second input from the third quantum node to the second quantum node; performing, at the second quantum node, a third quantum computation based on the encoding of the second input and the second component of the model, wherein performing the third quantum computation generates a third output;transmitting, via the second quantum communication channel, an encoding of the third output from the second quantum node to the third quantum node; and performing, at the third quantum node, a fourth quantum computation based on the encoding of the third output and a third component of the model implemented at the third quantum node, wherein performing the fourth quantum computation generates a fifth output.

27. A distributed quantum computing system (DQCS), comprising: a set of quantum nodes that includes a first quantum node and a second quantum node; a quantum communication network that communicatively couples the quantum nodes of the set of quantum nodes one or more processors; one or more memory devices, the one or more memory devices storing computer-readable instructions that when executed by the one or more processors cause the one or more processors to perform operations for performing a distributed computation via a model, the operations comprising: receiving, at a first quantum node of the set of quantum nodes, a first input to the model; performing, at the first quantum node, a first quantum computation based on the first input and a first component of the model that is implemented by the first quantum node, wherein performing the first quantum computation includes generating an encoding of a first intermediate output of the model; receiving, at a second quantum node of the set of nodes, a second input to the model; performing, at the second quantum node, a second quantum computation based on the second input and a second component of the model that is implemented by the second quantum node, wherein performing the second quantum computation includes generating an encoding of a second intermediate output of the model; transmitting, via the quantum communication network, the encoding of the first intermediate output from the first quantum node to a third quantum node of the set of quantum nodes; transmitting, via the quantum communication network, the encoding of the second intermediate output from the second quantum node to the third quantum node; determining, at the third quantum node, the first intermediate output based on the encoding of the first intermediate output and a quantum tomography algorithm;determining, at the third quantum node, the second intermediate output based on the encoding of the second intermediate output and the quantum tomography algorithm; and calculating, at the third quantum node, a result of the distributed computation 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 of claim 27, wherein the quantum tomography algorithm is a shadow quantum tomography algorithm.

29. A method comprising one or more aspects of any of the embodiments provided herein, or variants thereof.

30. A distributed quantum computing system (DQCS) comprising one or more aspects of any of the embodiments provided herein, or variants thereof.

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