Machine learning method and machine learning device using quantum systems

Abstract

A method for machine learning using a quantum system is provided. The method involves computing a set of primary qubits (5101, 5102, 5103, 5104, 5201, 5202, 5203, 515) ij ) and a set of auxiliary qubits (550 ij ), a neural network is associated with the quantum system. The neural network includes a first set of nodes (101, 102, 103, 104) and a second set of nodes (201, 202, 203). The neural network includes a plurality of connections (15) between the first set of nodes and the second set of nodes. ij ), each connection having a weight. The neural network includes a training data set including multiple configurations of a first set of nodes. Each connection (15 ij ) is the primary qubit (515 ij ) and ancillary qubits (550 ij ). The method includes initializing a quantum system to an initial quantum state. The method includes, for each configuration of a number of configurations obtained from a training data set, encoding the configuration onto input qubits that form a subset of the set of primary qubits. The method includes, for each configuration of the number of configurations, performing N operations, each operation including applying a sequence of unitary operators acting on at least a portion of the quantum system and performing a measurement of at least a portion of the quantum system. The method includes performing a measurement of one or more ancillary qubits. The method includes determining a weight for at least one of the connections of the plurality of connections based on the measurement of the at least one or more ancillary qubits.

Description

[Technical field]

[0001] FIELD OF THE DISCLOSURE Embodiments described herein relate to apparatus and methods for performing quantum computing, and more particularly, to apparatus and methods for computing solutions to machine learning problems using quantum systems, and more particularly quantum systems that include a plurality of quantum bits (qubits). [Background technology]

[0002] A neural network is a computational system used in the field of artificial intelligence to solve difficult computational tasks. Neural networks are inspired by biological neurons, e.g., neurons in animal brains. Neural networks are an example of machine learning systems, which can change as a program or computational system learns to solve a problem. With each example provided to a neural network, the neural network can be trained and improved.

[0003] A neural network includes nodes and connections between the nodes. A neural network can be designed to include several layers, with each layer including a respective set of nodes. There may be connections between nodes in adjacent layers, while there may be no connections between nodes in non-adjacent layers. In many cases, the number of nodes, layers, and connections can be very large.

[0004] Each connection in a neural network has a weight, e.g., a real number associated with the connection in question. A weight can be viewed as the strength of an interaction or signal between two nodes. The weights may be unknown a priori, and the task of the machine learning problem may be to determine or learn appropriate weights. Often, the neural network may be subject to additional conditions or constraints, and the task of the machine learning problem may be to determine appropriate weights for the connections such that a maximum amount of these conditions are at least approximately met.

[0005] The task of computing neural network weights can be extremely complex and time-consuming, especially as the size of the neural network grows. Several methods have been proposed to solve machine learning problems, but for some problems, even the best available methods can result in exponentially longer running times, making these methods intractable for large neural networks.

[0006] Therefore, there is a need for improved methods and apparatus for performing machine learning. Summary of the Invention

[0007] According to one embodiment, a method of machine learning using a quantum system is provided. The method includes providing a quantum system comprising a set of primary qubits and a set of ancillary qubits. A neural network is associated with the quantum system. The neural network includes a first set of nodes and a second set of nodes. The neural network includes a plurality of connections between the first set of nodes and the second set of nodes, each connection having a weight. The neural network includes a training data set comprising a plurality of configurations of the first set of nodes. Each connection of the plurality of connections is associated with both a primary qubit and an ancillary qubit. The method includes initializing the quantum system to an initial quantum state. The method includes, for each configuration of the number of configurations obtained from the training data set, encoding the configuration onto input qubits that form a subset of the set of primary qubits. The method includes performing N operations for each configuration of the number of configurations, each operation including applying a sequence of unitary operators acting on at least a portion of the quantum system and performing a measurement of at least a portion of the quantum system. The method includes performing a measurement of one or more ancillary qubits. The method includes determining a weight for at least one of the multiple connections based on a measurement of at least one or more ancillary qubits.

[0008] According to a further embodiment, there is provided a method of machine learning using a quantum system. The method includes providing a quantum system comprising a set of primary qubits and a set of ancillary qubits. A neural network is associated with the quantum system. The neural network includes a first set of nodes, a second set of nodes, and a plurality of connections between the first set of nodes and the second set of nodes, each connection having a weight. The neural network includes a teacher data set comprising a plurality of configurations of the first set of nodes. Each connection of the plurality of connections is associated with both a primary qubit and an ancillary qubit. The method further includes initializing the quantum system to an initial quantum state. The method further includes, for each configuration in the set sampled from the teacher data set, the following steps: · Encoding the configuration with input qubits that form a subset of the set of primary qubits. performing a plurality of first rounds of operations, where each first round includes (a1) applying a sequence of unitary operators selected from a first family of unitary operators, and (a2) performing a measurement of at least a portion of the quantum system. performing a plurality of second rounds of operations, where each second round includes (b1) applying a sequence of unitary operators selected from a second family of unitary operators, where the second family of unitary operators is different from the first family of unitary operators, and (b2) performing a measurement of at least a portion of the quantum system. The method further includes performing measurements of the one or more ancillary qubits and determining a weight for at least one of the connections of the plurality of connections based on the measurements of the at least one or more ancillary qubits.

[0009] According to a further embodiment, there is provided an apparatus for performing machine learning using a quantum system. The apparatus includes a quantum system including a set of primary qubits and a set of ancillary qubits as described herein. A neural network is associated with the quantum system. The neural network includes a first set of nodes, a second set of nodes, and a plurality of connections between the first set of nodes and the second set of nodes, each connection having a weight. The neural network includes a teacher data set including a plurality of configurations of the first set of nodes. Each connection of the plurality of connections is associated with both a primary qubit and an ancillary qubit. The apparatus includes a quantum processing device for evolving the quantum system according to one or more short-range unitary operators. The apparatus includes an encoding device for interacting with input qubits forming a subset of the set of primary qubits. The apparatus includes a measurement device for measuring at least a portion of the quantum system. The apparatus is configured to initialize the quantum system to an initial quantum state. The apparatus is configured to perform the following steps for each configuration of the plurality of configurations obtained from the teacher data set: - encoding a configuration in the input qubits by interacting with the input qubits by an encoding device. - performing N operations, each of which includes applying, by a quantum processing device, a sequence of unitary operators acting on at least a portion of the quantum system, and performing, by a measurement device, a measurement of at least a portion of the quantum system. The apparatus is configured to perform, with a measurement device, measurements of one or more ancillary qubits. The apparatus is configured to determine a weight for at least one of the connections of the plurality of connections based on the measurements of the at least one or more ancillary qubits.

[0010] Embodiments are also directed to methods for operating the disclosed systems and apparatus, as well as to the use of the disclosed systems to perform methods according to embodiments described herein.

[0011] Further advantages, features, aspects and details which can be combined with the embodiments described herein are evident from the dependent claims, the description and the drawings. [Brief description of the drawings]

[0012] A full and enabling disclosure, to one of ordinary skill in the art, is set forth more particularly in the remainder of the specification, including reference to the accompanying drawings, in which: [Figure 1a] 1 illustrates an example of a neural network as described herein. [Figure 1b] 1 illustrates an example of a neural network as described herein. [Diagram 2] 1 illustrates an example of a neural network as described herein. [Diagram 3] 1 shows an example of a classical spin model relevant to the neural networks described herein. [Figure 4] 1 shows an example of a configuration of training data for the neural network described in this specification. [Diagram 5] 1 illustrates an example of a quantum system related to the neural networks described herein. [Figure 6] 1 illustrates an example of a quantum system having a target state for a set of data qubits as described herein. [Figure 7] 1 illustrates an example of a quantum system related to the neural networks described herein. [Figure 8] 1 illustrates an example of a quantum system related to the neural networks described herein. [Figure 9] 1 illustrates an example of an apparatus for performing machine learning using a quantum system according to embodiments described herein. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0013] Reference will now be made in detail to various exemplary embodiments, one or more examples of which are illustrated in the figures, each example being provided by way of illustration and not by way of limitation. For example, features illustrated or described as part of one embodiment can be used on or in conjunction with other embodiments to yield still further embodiments. The present disclosure is intended to include such modifications and variations.

[0014] In the following description of the drawings, the same reference numbers refer to the same components. Generally, only the differences with respect to individual embodiments are described. The structures shown in the drawings are not necessarily drawn to scale and may include details that are drawn in an exaggerated manner to allow a better understanding of the embodiments.

[0015] The embodiments described herein include a method for performing machine learning using a quantum system. A quantum processing device can be used to calculate unknown weights of a neural network, i.e., to train the neural network. For clarity and without intending to be limiting in scope, an exemplary embodiment of the method is described below. The embodiment is then described in detail.

[0016] FIG. 1b illustrates an exemplary neural network that includes a visible layer (first set of nodes) and a hidden layer (second set of nodes). The visible layer illustrated in FIG. 1b is composed of nodes 101, 102, 103, and 104. In other examples, the visible layer may include more or fewer nodes. The hidden layer illustrated in FIG. 1b is composed of nodes 201, 202, and 203. In other examples, the hidden layer may include more or fewer nodes.

[0017] A neural network contains multiple connections between the visible and hidden layers. In Figure 1b, every node in the visible layer is connected to every node in the hidden layer. In other words, the neural network is fully connected. There are a total of 4 × 3 = 12 connections, and the number of connections is 15. ijwhere i ranges from 1 to 4 and j ranges from 1 to 3. There are no connections between nodes in the visible layer and no connections between nodes in the hidden layer.

[0018] The neural network shown in FIG. 1b includes one hidden layer. In other examples, the neural network may include two or more hidden layers (deep neural network). In addition, the neural network may include source nodes and / or sink nodes. A source node is a node that does not belong to any visible layer or hidden layer, but is connected to some or all of the nodes in the visible layer. A sink node is a node that does not belong to any visible layer or hidden layer, but is connected to some or all of the nodes in the hidden layer.

[0019] Each connection between the i-th node and the j-th node in the neural network shown in Fig. 1b is ij has weight. Connection 15 ij The weight of is w ij The weight w ij At least some of the weights w are unknown a priori, e.g., taken from the set {0,1}. The goal of a machine learning problem is to estimate the unknown weights w ij The objective of the present invention is to determine or learn.

[0020] As shown in FIG. 3, the neural network of FIG. 1b can be associated with a classical spin model. As shown in FIG. 3, a classical spin is assigned to each node of the neural network. Thus, the classical spin model shown in FIG. 3 includes classical spins s1, s2, s3, s4 located at nodes 101, 102, 103, 104, respectively, in the visible layer, and classical spins s5, s6, s7 located at nodes 201, 202, 203, respectively, in the hidden layer. Each classical spin can be a two-state spin, i.e., a spin that can take two possible values ​​or states. For example, each spin si can belong to the set {1,-1}. The classical spin model can be an Ising spin model with an energy function of the form: E(s1,s2,s3, )=-Σa i s i -Σb j s j -Σw ij s i s j Here, the index i spans all nodes in the visible layer (i.e., i=1,2,3,4), and the index j spans all nodes in the hidden layer (i.e., j=1,2,3). ij is the weight of the connection between the i-th node and the j-th node, as mentioned above. Therefore, the interaction between spins in the classical spin model is determined by the weight of the corresponding connection in the neural network. Also, a i , b j are the coefficients (representing the local fields of the spin model). The Boltzmann distribution for the classical spin model is given by P SM (s)=exp(-E(s1,s2,s3,...)) / Z where Z is the partition function of the classical spin model and exp is the exponential function. Considering the above, the neural network shown in FIG. 1b is associated with or mapped onto the classical spin model (the Ising model), and the energy function of the classical spin model, and therefore the Boltzmann distribution, depends on the weights of the neural network.

[0021] The neural network may further include a training data set as an input. The training data set consists of a number of known configurations, referred to herein as training data structures. Each training data structure assigns a particular state or value to each spin located at a node in the visible layer. In the above example, each training data structure has the form t=(t1, t2, t3, t4), where t i is node 10 in the visible layer i , and each t i is equal to 1 or -1. This is shown in Figure 4. For example, for the neural network shown in Figure 1b, an example training data set is as follows: {(1,1,1,-1),(-1,-1,1,1),(-1,1,-1,1),(-1,1,1,1,1),(-1,-1,1-1,-1)} It consists of five training data structures: The training data set can be stored in a classical computing system.

[0022] Each training data configuration t of the neural network has a certain probability P TD (t). That is, a probability distribution can be defined for the training data set. For example, the training data probability distribution can be a uniform probability distribution, where each training data configuration has the same probability, i.e., P TD It occurs with (t) = constant. The probability distribution of the training data is part of the input to the machine learning problem, i.e., this distribution is known.

[0023] The first probability distribution for the training data set is the training data probability distribution P TD In addition to the above, we can consider a second probability distribution for the training data set, namely the Boltzmann distribution P SM We can consider a second probability distribution over the training data set by considering the marginal distribution of . For each training data configuration t, P marg (t) denotes the probability of finding a classical spin model spin in a spin configuration such that the spin associated with the visible layer is in configuration t and the remaining spins can be in any state. The probability P marg (t) is the Boltzmann distribution P SM It is obtained by summing (integrating) all the spins located at nodes outside the visible layer, starting from

[0024] Machine learning problems have condition P TD (t)=P marg The weights w for the connections of the neural network are calculated so that (t) is satisfied (at least approximately) for all configurations t of training data in the training data set. ijIn other words, this condition indicates that the probability distribution of the training data is approximately equal to the marginal distribution of the Boltzmann distribution.

[0025] The approach taken by the present disclosure to solve machine learning problems is to use a quantum system onto which a neural network is mapped. As will be described in more detail below, the method according to the present invention uses a quantum system in which each weight of the neural network is associated with its own quantum degree of freedom. That is, the quantum system includes, for each weight, a respective qubit (an "ancillary qubit"). The method includes performing an iterative sequence of operations (unitary operations and measurements) on the quantum system such that the quantum states of the ancillary qubits converge to a final state, and the unknown weights of the neural network can be read out by performing a final measurement at the end of the quantum computation.

[0026] The mapping from a neural network to a quantum system is described below. Figure 5 shows a quantum system associated with the neural network shown in Figure 1b. The quantum system shown in Figure 5 includes a set of primary qubits arranged along a portion of a first two-dimensional lattice. The set of primary qubits includes primary qubits 5101, 5102, 5103 and 5104, primary qubits 5201, 5202 and 5203, and primary qubit 515. ij where i ranges from 1 to 4 and j ranges from 1 to 3. In other words, the set of primary qubits includes three subsets of primary qubits.

[0027] Primary qubits 5101, 5102, 5103 and 5104, also called data qubits, are associated with the visible layer of the neural network. i is the primary qubit 510 shown in FIG. i Associated with Node 10 i Classical spin s located at i The possible states of the associated primary qubit 510i In one example, qubit 510 is encoded as one of the possible quantum (basis) states. i The quantum state |0> of i = 1, and the quantum state |1> is the spin state s i =-1. As explained in more detail below, data qubits are optional, i.e., they can be omitted.

[0028] The primary qubits 5201, 5202 and 5203 are associated with a hidden layer of the neural network. Each node 20 of the hidden layer shown in FIG. j is the primary qubit 520 shown in FIG. j Similar to the data qubits, node 20 j Classical spin s located at i The possible states of the associated primary qubit 520 j is encoded as the possible quantum (ground) states of

[0029] Furthermore, the primary qubit 515 ij is associated with a connection in the neural network. i and node 20 j 15 for each connection between ij is the primary qubit 515 shown in FIG. ij Associated with Node 10 i The classical spin s located at i and node 10 j The classical spin s located at j The possible parity of the interaction between the associated primary qubits515 ij In one example, the qubit 515 ij The quantum state |0> of i ands j The quantum state |1> encodes the situation where spin s and s are aligned with each other (i.e., both spins are equal to 1, or both spins are equal to -1, i.e., the parity is even). i ands jand are anti-aligned with respect to each other (i.e., one spin is equal to 1 and the other spin is equal to -1, i.e., the parity is odd).

[0030] In addition to the set of primary qubits, the quantum system may include ancillary qubits 550, which may be located along a portion of a second two-dimensional lattice, as shown in FIG. ij The first and second two-dimensional lattices may be arranged such that each ancillary qubit is paired with, i.e., adjacent to, a primary qubit.

[0031] The ancillary qubits are associated with the weights of the connections of the neural network, i.e., node 10 shown in FIG. i and node 20 j 15 for each connection between ij is the ancillary qubit 550 shown in FIG. ij Associated with the auxiliary qubit 550 ij is the weight of the connection w ij In one example, the ancillary qubit 550 ij The quantum state |0> of has weight w ij = 0, and the quantum state |1> has weight w ij Code =1.

[0032] Considering the above, each connection in a neural network is associated with both a primary qubit and an ancillary qubit, which encodes the weight of the connection.

[0033] As noted above, the approach adopted by the present disclosure is to associate ancillary qubits with a quantum state Ψ that encodes a solution to a machine learning problem. solution >, that is, to execute a sequence of quantum operations that drives the state Ψ solution > encodes the unknown weights of the neural network. For this purpose, several possible sequences of quantum operations are possible, but one particular example is described below as an illustration of the method described herein.

[0034] The method can begin by initializing each qubit in the system (i.e., the primary qubit and the ancillary qubit) to an initial quantum state, e.g., state |+>=(|0>+|1>) / √2. This can be done, for example, by declaring the quantum system to have a Hamiltonian Σ k σ x (k) This can be achieved by cooling the system to the ground state of σ x (k) is the Pauli operator ("sigma-x") acting on the kth qubit, and the summation is over all qubits in the quantum system.

[0035] Furthermore, a first training data configuration t1 is obtained from the training data set. The first training data configuration t1 is a training data probability distribution P TD For example, P TD If x is a uniform distribution, the first training data configuration may be obtained by generating a uniform random sample of the training data set.

[0036] A number of quantum operations are performed in association with the first teaching configuration t1, some of which operate on primary qubits, some of which operate on ancillary qubits, and some of which operate on both at least one primary qubit and at least one ancillary qubit. The purpose of the quantum operations associated with the first teaching configuration t1 is to "write" the configuration t1 into the quantum system and to generate a quantum state Ψ that encodes a solution to the machine learning problem. solution > (at this early stage, Ψ solution After the quantum operations related to the first training data configuration t1 have been performed, the distribution P TD A second training data configuration t2 is obtained from the training data set by randomly sampling t. A number of other quantum operations are performed in association with the second training data configuration t2. These quantum operations convert the ancillary qubits into a quantum state Ψ that encodes the solution.solution >. This procedure continues: at each step of the procedure, a training data configuration is obtained by sampling the training data set, this training data configuration is written into the quantum system, and the quantum system is put into the quantum state Ψ solution To further develop the quantum state towards Ψ, multiple quantum operations are performed. By repeating this procedure many times, the ancillary qubits are at least in quantum state Ψ solution > converges to a quantum state close to

[0037] Returning to the first training data configuration t1, the quantum operations performed in relation to t1 may include the following operations:

[0038] First, one or more strong fields (e.g., magnetic fields) are applied to bias data qubits (e.g., primary qubits 5101, 5102, 5103, and 5104 in FIG. 5) toward a target quantum state that encodes a first teacher data configuration t1. t1=(t 11 ,t 12 ,t 13 ,t 14 ) (where node 10 i The value associated with t 1i is equal to 1 or -1), which means that, for example, t 1i qubit 510 when is equal to 1 i Apply a field that biases t towards the state |0>, and 1i qubit 510 when is equal to -1 i This can be achieved by applying a field that biases |1> towards the state |1>. The strong field therefore encodes or “writes” the information contained in the first teacher data configuration t1 into the set of primary qubits.

[0039] Alternatively, the teacher data configuration may be encoded in the primary qubits in other ways, for example by two-body interactions between pairs of connected primary qubits as described herein, in which case no data qubits need be present, as will be described in more detail below.

[0040] Next, a sequence of unitary operators having the form A1B1A2B2... is applied to the quantum system. k and B. k The number of operators can become very large, and a sequence may contain hundreds or thousands of operators. k and B. k can operate entirely within the set of primary qubits. k and B. k can be obtained from the first family of unitary operations. For example, for each A k can be an operator of the form:

number

number

[0041] The parameters β and Ω are considered as variational parameters, as explained below.

[0042] After the sequence A 1 B 1 A 2 B 2 ··· is applied to the quantum system, an energy measurement is performed, i.e., a measurement of the following Hamiltonian (referred to herein as the total Hamiltonian):

number

[0043] After the total Hamiltonian has been measured, the measured energy is recorded. The qubits may then be reinitialized (e.g., in the state |+> above), one or more fields are applied again, and the first teacher configuration t1 is written to the data qubits (alternatively, the fields in question may be switched on continuously, i.e., never switched off in the first place). Another sequence A1B1A2B2··· is applied to the quantum system, setting new values ​​for the parameters β and Ω. The new values ​​of β and Ω can be obtained by starting from the previously obtained values ​​of β and Ω and adding small perturbations δβ and δΩ, respectively. After the sequence A1B1A2B2···, another measurement of the total Hamiltonian is made. The measured energy of the later measurement is compared to the previously measured energy. The values ​​of β and Ω corresponding to the smaller of the two energies are kept, and the values ​​corresponding to the larger energy are discarded. The kept values ​​of β and Ω become the starting point for the next round of operations. The process continues. That is, at each round, a sequence A1B1A2B2··· is applied to the quantum system. The values ​​of β and Ω used in the sequence are determined by starting from the retained values ​​of the previous energy comparison and applying small perturbations to these retained values. This iterative procedure is repeated many times. The aim of this process is to lower the energy as much as possible, i.e. to find values ​​of β and Ω that correspond to quantum states with low energy. The operations A1B1A2B2··· can be performed while one or more fields are applied to bias the data qubits towards a target state that encodes the first teacher data configuration t1. In doing so, the quantum system is driven to a low-energy quantum state subject to the condition that the data qubits are biased towards the target state in question. This can be understood as a minimum energy search under side conditions that keep part of the system in a fixed state. The fixed part of the quantum system acts as an effective local field on the dynamic part. After a given number of iterations (e.g., 100, 1000 or more iterations), the resulting optimal values ​​β1 and Ω1 are recorded.

[0044] Afterwards, the qubits are again reinitialized (e.g., in state |+>) and the first teacher data configuration t1 is again encoded into the data qubits. Furthermore, the sequence A1B1A2B2... corresponding to the previously obtained optimal values ​​β1 and Ω1 is applied to the quantum system. The resulting quantum state is denoted |Ψ1>. In the latter state, the alternating sequence C1D1C2D2... is applied. The number of alternations may again be in the hundreds or thousands. The operator C k and D. k may be operators taken from a second family of unitary operators different from the first family of operators. For example, each operator C i may have the following form:

number

number

[0045] Here, the index k ranges over all connections in the neural network, and (k,p) and (k,a) are as defined above. Note that τ is a parameter. The parameters α and τ play the role of variational parameters, as do the parameters β and Ω. In particular, the operator U a (τ) couples the primary qubit to the auxiliary qubit, so that information can be “moved” from the primary qubit to the auxiliary qubit.

[0046] After the sequence C1D1C2D2··· is applied to the state |Ψ1〉, the total Hamiltonian is measured and the measured energy is recorded. As described above, multiple rounds of this kind are performed to minimize the energy. In each round, the sequence C1D1C2D2··· is applied to the state |Ψ1〉 using appropriate values ​​of parameters α and τ, and the total Hamiltonian is then measured. The values ​​of α and τ selected for a particular round are based on the previously obtained values ​​of α and τ corresponding to the smallest measured energy, with small perturbations added. After a given number of iterations (e.g., 100, 1000 or more iterations), the resulting optimal values ​​α1 and τ1 are recorded. As described above, this can be an energy minimization subject to the condition that the data qubits are biased towards a target state that encodes the first teacher data configuration t1.

[0047] Thus, at this stage of the procedure, the optimal values ​​β1, Ω1 and τ1 have been obtained in relation to the first training data configuration t1.

[0048] The above procedure is performed multiple times in relation to multiple training data configurations. TD By sampling, a second teacher data configuration t2, a third teacher data configuration t3, ... are generated. Such a teacher data configuration t i The above-mentioned operations are performed for each configuration t i For this, the optimal value β i , Ω i is determined by repeatedly applying the sequence A1B1A2B2..., followed by measuring the total Hamiltonian. Similar to the state |Ψ1> considered above for the first training data configuration t1, |Ψ i > for the i-th training data configuration t i The optimal value β related to i and Ω i Let A1B1A2B2... denote the quantum state obtained by applying the sequence A1B1A2B2... corresponding to t1. Similarly to the procedure above for t1, multiple sequences C1D1C2D2... denote the quantum state obtained by applying the sequence A1B1A2B2... corresponding to t1. i>, each followed by a measurement of the total Hamiltonian, and finally the configuration t i The optimal value α related to i and τ i In particular, t i When applying the first sequence C1D1C2D2... in relation to the previous training data configuration t i-1 The optimal value α obtained in i-1 and τ i-1 It may be useful to start from the number of operators in the sequence C1D1C2D2..., i.e., the configuration t i The length of the sequence considered in relation to the previous training data configuration t i-1 The optimal α found in relation to i-1 and τ i-1 In order not to deviate too far from α, it can be kept relatively small (e.g., a sequence may contain no more than 100 operators). In other words, the value α i-1 and τ i-1 In order to avoid losing the information contained in i-1 and τ i-1 The optimal α in a small area around i and τ i It may be useful to look for

[0049] After performing the above iterative procedure on a number of training data configurations sampled from the training data set, an auxiliary qubit (e.g., qubit 550 shown in FIG. 5) is ij ) are measured in the basis {|0>,|1>}. Measurement of each ancillary qubit results in a numerical value equal to 0 or 1. If the measurement result of an ancillary qubit is 0, the associated weight of the neural network is determined to be equal to 0. Similarly, if the measurement result is 1, the weight is determined to be equal to 1. The initial unknown weights of the neural network are determined by reading out the ancillary qubits. In particular, the weights determined by the methods according to embodiments described herein satisfy the condition P marg (t) ≒ P TD(t) is satisfied for each training data configuration t in the training data set. Thus, the embodiments described herein provide a method for training a neural network using a quantum system.

[0050] In view of the above, according to one embodiment, a method for machine learning using a quantum system is provided. The method includes providing a quantum system including a set of primary qubits and a set of ancillary qubits. A neural network is associated with the quantum system. The neural network includes a first set of nodes and a second set of nodes. The neural network includes a plurality of connections between the first set of nodes and the second set of nodes, each connection having a weight. At least some of the weights may be unknown. The neural network includes a teacher data set with a plurality of configurations of the first set of nodes. Each connection of the plurality of connections is associated with both a primary qubit and an ancillary qubit. The method includes initializing the quantum system to an initial quantum state. The method includes, for each configuration of the plurality of configurations obtained from the teacher data set, encoding the configuration onto input qubits that form a subset of the set of primary qubits. The method includes performing N operations for each configuration of the plurality of configurations, each operation including applying a sequence of unitary operators acting on at least a portion of the quantum system and performing a measurement of at least a portion of the quantum system. The method includes performing a measurement of one or more ancillary qubits. The method includes determining a weight for at least one of the multiple connections based on a measurement of at least one or more ancillary qubits.

[0051] Thus, the embodiments described herein allow for the use of quantum systems to determine solutions to machine learning problems. Compared to determining solutions to machine learning problems using only classical computing systems, i.e., without using quantum systems, the embodiments described herein can reduce the computation time required to solve the machine learning problem, or make the machine learning problem entirely computable. In other words, compared to classical computing systems, the embodiments described herein may allow for the machine learning problem to be solved faster, or even to find such a solution in the first place, since it would take too long to compute such a solution with classical computing systems.

[0052] Furthermore, the auxiliary qubits of the quantum system are configured to code the weights of the connections. In other words, by providing auxiliary qubits, the embodiments described herein associate a quantum degree of freedom with each weight of the neural network, i.e., the weights themselves are "quantized". This differs from approaches using quantum systems to solve machine learning problems, where the weights are associated with classical degrees of freedom (e.g., each weight can be associated with a magnetic field acting on the primary qubit without the presence of auxiliary qubits). Quantizing the weights allows the full benefit of quantum mechanics to be exploited to determine the unknown weights. That is, it is believed that the auxiliary qubits associated with the weights can exhibit quantum mechanical effects (e.g., they can be in a superposition state that includes a combination of multiple weights), and such quantum effects can be exploited to significantly reduce the time required to determine the unknown weights, i.e., to speed up the computation of the weights. In comparison, systems in which neural networks are associated with quantum systems, but where the weights are encoded as (semi-)classical degrees of freedom such as magnetic fields, may at best only exhibit a partial speedup in computation time. The speedup achieved by such a system is likely to be smaller than that achievable by current methods, since the latter involve quantization of the weights themselves, i.e., a computation that is fully quantum in nature.

[0053] Neural Networks

[0054] The embodiments described herein include the concept of a neural network. Figures 1a and 1b show examples of neural networks.

[0055] A neural network includes a plurality of nodes. The plurality of nodes may include a first set of nodes, i.e., a "visible layer," such as nodes 101, 102, 103, and 104 shown in FIG. 1a. The plurality of nodes may include a second set of nodes, i.e., a "hidden layer," such as nodes 201, 202, and 203 shown in FIG. 1a. The plurality of nodes may include a third set of nodes, a fourth set of nodes, etc. (not shown in FIG. 1a), each of which may be a further "hidden layer." That is, a neural network may include multiple hidden layers. When more than one hidden layer is provided, the neural network may be referred to as a deep neural network.

[0056] The neural network may include further nodes, for example, a neural network may include a source node 100 and / or a sink node 200 as shown in Figure 1a.

[0057] The neural networks shown in the figures are small neural networks for ease of representation. Much larger neural networks are also contemplated. For example, a neural network may include 1000 nodes, or up to 100 million nodes, and / or may include 10 layers, or up to 100 layers.

[0058] A neural network includes a number of connections between a first set of nodes and a second set of nodes. For example, FIG. 1a shows 15 connections between the first set of nodes and the second set of nodes. ij Indicates connection 15 ij is node 10 of the first set of nodes i and node 20 of the second set of nodesj There are no connections within the first set of nodes. There are no connections within the second set of nodes. There are no connections between any set of nodes that belong to a layer.

[0059] The neural network may include one or more connections between a source node and a first set of nodes, for example as shown in Figure 1a. Additionally or alternatively, the neural network may include one or more connections between a sink node and a second set of nodes, for example as shown in Figure 1a.

[0060] Figure 1b shows a further example of a neural network. Similar to the neural network shown in Figure 1a, the neural network of Figure 1b includes nodes 101, 102, 103 and 104 forming a first set of nodes (visible layer) and nodes 201, 202 and 203 forming a second set of nodes (hidden layer). The neural network may optionally include source and sink nodes (not shown). The neural network shown in Figure 1b is fully connected. Each node of the first set of nodes is connected to each node of the second set of nodes. Thus, the neural network of Figure 1b has connections 15 ij where i ranges from 1 to 4 and j ranges from 1 to 3, i.e., 4×3=12 connections in total.

[0061] Fig. 2 shows a further example of a neural network. This neural network is a deep neural network. The neural network comprises nodes 210 forming a first set of nodes (visible layer) and nodes 220 forming a second set of nodes (hidden layer). The neural network comprises connections 215 between the first set of nodes and the second set of nodes. The neural network comprises nodes 230 forming a third set of nodes, nodes 240 forming a fourth set of nodes and nodes 250 forming a fifth set of nodes. The third set of nodes, the fourth set of nodes and the fifth set of nodes can be understood as further hidden layers of the neural network in addition to the nodes 220. The neural network comprises connections 225 between the second set of nodes and the third set of nodes, connections 235 between the third set of nodes and the fourth set of nodes and connections 245 between the fourth set of nodes and the fifth set of nodes.

[0062] The neural network may be a deep neural network. The neural network may include a third set of nodes. The neural network may include a number of connections between the second set of nodes and the third set of nodes, such as the connections 225 shown in FIG. 2. There are no connections within the third set of nodes. There are no connections between the first set of nodes and the third set of nodes.

[0063] The neural network may include a fourth set of nodes. The neural network may include a number of connections between the third set of nodes and the fourth set of nodes, such as connection 235 shown in FIG. 2. There are no connections within the fourth set of nodes. There are no connections between the first set of nodes and the fourth set of nodes. There are no connections between the second set of nodes and the fourth set of nodes.

[0064] Thus, a deep neural network may include multiple sets of nodes (layers), and for each k, there may be one or more connections between the kth set of nodes and the (k+1)th set of nodes. Also, there may be one or more connections between the kth set of nodes and the (k-1)th set of nodes. However, there are no connections between the kth set of nodes and any other set of nodes; that is, connections exist only between adjacent sets of nodes. Furthermore, there are no connections within the kth set of nodes.

[0065] The neural network shown in Figure 2 is fully bipartitely connected: every node in the kth set of nodes is connected to every node in the (k+1)th set of nodes.

[0066] The connections in a neural network may have weights. A weight can be considered as the strength of an interaction or signal between two nodes. The weight of a connection between the i-th node and the j-th node is w ij For example, in Figures 1a and 1b, all connections w ij Weight w ij A weight can be a numerical value, such as 0 or 1 (or, equivalently, 1 or -1). Other weights are possible, for example, the weight can be any numerical value (integers or real numbers).

[0067] A connection between two nodes with weight 0 can be understood as no connection exists between the two nodes. Thus, without loss of generality, a neural network can be thought of as being "fully bipartitely connected"; that is, every node in a first set of nodes is connected to every node in a second set of nodes, every node in the second set of nodes is connected to every node in a third set of nodes, and so on. Some of the connections may have a weight of zero. Examples of fully bipartitely connected neural networks are shown in Figures 1b and 2.

[0068] In terms of machine learning, at least some of the weights of a neural network may be unknown weights. The objective may be to determine or "learn" one or more unknown weights of a neural network. The notion of "machine learning problem" is used herein to refer to a task that involves the determination of one or more unknown weights of a neural network, i.e., the task of training a neural network. In particular, the machine learning problem described herein may be an unsupervised learning problem, including the task of unsupervised training of a neural network.

[0069] The neural network can be associated with a classical spin model, which may include multiple classical spins s1, s2, s3,... Each classical spin s i may be assigned to or located at each i-th node of a neural network. A classical spin may contain a value or a state. A classical spin may have a value taken from a binary set or may be in one of two states. For example, each classical spin is mathematically denoted by s i ∈{1,-1}. It should be understood that the values ​​may be different. For example, the value or state may be s i ∈{0,1}.

[0070] In some embodiments, the classical spin model is a classical spin s for each node of the neural network. i In other embodiments, one or more nodes of the neural network may include a classical spin s i may not be assigned.

[0071] For example, a classical spin may be associated with the source node and / or a further classical spin may be associated with the sink node. Alternatively, the classical spin model may not include classical spins associated with the source node or the sink node. The spin model may include classical spins associated only with visible and hidden layer nodes of the neural network. In further embodiments, the classical spin associated with the source node and / or the classical spin associated with the sink node may be fixed spins. For example, these spins may both be fixed to a value of 1.

[0072] For example, Figure 3 shows the neural network of Figure 1b with classical spins s1, s2, s3 and s4 located at four nodes in the visible layer, and classical spins s5, s6 and s7 located at three nodes in the hidden layer. If there are further hidden layers (deep neural networks), further spins can be placed at the nodes of these further hidden layers as well.

[0073] In the classical spin model, we refer to the probability distribution P SM Each configuration s = (s1,s2,s3,...) of classical spins in the classical spin model can be given a probability distribution P SM This has a certain probability of occurring according to

[0074] In some embodiments, the classical spin model has an energy function E(s1,s2,s3,...) that assigns an energy to each configuration s=(s1,s2,s3,...) of classical spins. For example, if there is a connection (particularly a connection with a non-zero weight) between the i-th node and the j-th node of the neural network, then the spins located at the i-th and j-th nodes are coupled to each other in the classical spin model. Thus, the energy function E(...) may include a respective interaction term for each connection of the neural network.

[0075] Probability distribution P SMmay be the Boltzmann distribution of the classical spin model in question. The probability of occurrence of a spin configuration s = (s1,s2,s3,...) is given by the formula P SM It is given by (s)=exp(-E(s1,s2,s3,···)) / Z, where Z is the partition function of the classical spin model and exp(·) is the exponential function. The neural network can be a restricted Boltzmann machine.

[0076] Probability distribution P of the classical spin model SM may depend at least in part on the weights of the connections of the neural network. In particular, the energy function E(s1,s2,s3,...) may depend at least in part on the weights. For example, the classical spin model may be the Ising spin model. The energy function of the Ising spin model has the form E(s1,s2,s3, )=-Σa i s i -Σb j s j -Σw ij s i s j where index i spans all nodes in the first set of nodes (the "visible layer") and index j spans all nodes in the second set of nodes (the "hidden layer"). ij is the weight of the connection between the i-th node and the j-th node, and a i and b j are coefficients. For deep neural networks, an energy function of the Ising spin model may also be provided. For example, if the neural network includes a third set of nodes (such as node 230 shown in FIG. 2), i.e., a further hidden layer, the energy function may be -Σc k s k and / or -Σw jk s j s k where k spans all nodes in the third set of nodes and s k is the classical spin located at the kth node of the third set of nodes, and c k is the coefficient, and wjk is the weight of the connection between the jth node in the second set of nodes and the kth node in the third set of nodes.

[0077] The neural network may include a training data set comprising multiple configurations of a first set of nodes, i.e., the visible layer. The configurations in the training data set are also referred to herein as training data configurations. A training data configuration may be understood as an assignment of a particular value or state (e.g., 1 or -1) to each spin associated with the first set of nodes (i.e., the visible layer). In other words, a training data configuration is a set of values, each value being associated with a node of the first set of nodes. Each value included in the training data configuration may be an element of a set of possible states of a spin of a classical spin model. For example, if {1,-1} is a set of possible states of a classical spin model, then each value included in the training data configuration belongs to the set {1,-1}. A training data configuration may be considered as a partial configuration of spins, where spin states are assigned to nodes of the visible layer but not to the remaining nodes of the neural network.

[0078] For example, the first set of nodes may consist of nodes v1, v2, v3,.... The training data structure may have the form t = (t1, t2, t3,...), where each t i is a numerical value, e.g., each t i is either 1 or -1. For each i, the value t i is the node v in the first set of nodes i is associated with the value t i may be the same as the number of nodes in the first set of nodes, i.e., the number of nodes in the visible layer.

[0079] FIG. 4 shows the concept of the teacher data configuration of the neural network shown in FIG. 1b. In the neural network shown in FIG. 1b, the first set of nodes includes four nodes. Each teacher data configuration can have the form t=(t1, t2, t3, t4) as shown in FIG. 4, and each value t i is node 10 iAn example of the teacher data configuration may be t=(1, 1, 1, -1).

[0080] The training data set includes multiple training data configurations. For example, for the neural network shown in Figure 1b, an example of the training data set is as follows: {(1,1,1,-1),(-1,-1,1,1),(-1,1,-1,1),(-1,1,1,1,1),(-1,-1,1-1,-1)}

[0081] A training data set can be considered as the input to a machine learning problem, i.e., data known to the user. A training data set can be used to train a neural network.

[0082] In some embodiments, the teacher data configuration may be obtained by performing one or more computational processing operations on the initial data string. Such computational processing operations may be performed to obtain some or all of the teacher data configurations in the teacher data set. For example, the initial data string may be a bit string, where a bit (i.e., a value of 0 or 1) is assigned to each node in the visible layer. The initial data string may be processed by replacing each bit 0 with a value of 1 and each bit 1 with a value of -1. Thus, a training data configuration may be obtained in which the training data consists of a sequence of values, each value being 1 or -1. In another example, the initial data string consists of a sequence of values, each equal to 1 or -1, but the length of the sequence is shorter than the number of nodes in the visible layer. Such an initial data string may be processed by adding a number of values ​​equal to 1 to the end of the string to match the total number of nodes in the visible layer. The resulting string of values ​​may be the teacher data configuration, i.e., a number of values, each value associated with each node in the visible layer.

[0083] The neural network has a training data probability distribution P TDEach teacher data configuration occurs with a certain probability according to the probability distribution of the teacher data. For example, the probability distribution of the teacher data can be a uniform probability distribution in which each teacher data configuration occurs with the same probability. For example, if the teacher data set is composed of T teacher data configurations, the probability of each teacher data configuration t = (t1, t2, t3, ...) is 1 / T, or P TD (t) = 1 / T. It is important to understand that the uniform distribution is just one possible example of a probability distribution for the training data, and that non-uniform distributions can also be considered.

[0084] Probability distribution of training data P TD can be known. The probability distribution of the training data is provided as the input for the machine learning problem. In comparison, the probability distribution P SM can depend on unknown variables, i.e., the connection weights. Furthermore, the probability distribution of the training data can assign probabilities to configurations that include only visible layer nodes, whereas the probability distribution of the classical spin model can assign probabilities to configurations that also include hidden layer nodes.

[0085] As shown above, the teacher data configuration t = (t1, t2, t3, . . .) has a probability P TD (t) can be assigned to the probability distribution P SM Based on this, we can assign a second probability to the teacher configuration t = (t1, t2, t3, . . . ). This can be understood as follows: the teacher configuration t = (t1, t2, t3, . . . ) can be seen as a partial configuration of the classical spins of the classical spin model. This means that the teacher configuration has a probability of having a value t i This is because the training data structure is a probability distribution P SM We can consider the marginal probability of occurrence as a configuration of spins located on a first set of nodes according to: Mathematically, the marginal probability can be determined according to the following formula: P marg (t)=Σ u PSM (t,u) where each u = (u1, u2, u3, . . . ) is a classical spin u outside the first set of nodes. i where the tuple (t,u) forms a "complete" configuration of all spins in the classical spin model. Furthermore, a summation is performed over all such configurations u. That is, the configurations u are "summed up" to determine a marginal probability distribution for the teacher configuration t. This marginal probability distribution is the second probability assigned to the teacher configuration t.

[0086] A machine learning problem may involve the task of determining one or more unknown weights of a neural network. The weights of the neural network may be subject to certain conditions or constraints. For example, a possible condition for the weights may be: for each training data configuration t, the bounded probability P marg (t) is the probability P TD (t). Thus, the task of a machine learning problem is to determine the probability of P marg (t) ≒ P TD This may include determining appropriate weights for the neural network such that (t).

[0087] The neural network calculates the probability distribution of training data P TD The probability of the teacher data configuration according to is the probability distribution P SM The marginal probability distribution of P TD The condition may be that the probability of the training data being structured according to

[0088] Qubit System

[0089] Embodiments described herein include quantum systems having a plurality of qubits, including a set of primary qubits and a set of ancillary qubits.

[0090] A qubit as described herein is a quantum mechanical two-level system. A qubit may include two quantum basis states |0> and |1> that represent possible quantum states of the qubit. According to the superposition principle of quantum mechanics, all superpositions of the form a|0>+b|1> are possible quantum states of the qubit, where a and b are complex numbers. Mathematically, a qubit may be represented in a two-dimensional vector space. A plurality of qubits may have quantum basis states that correspond to configurations in which each qubit of the plurality of qubits is in either quantum state |0> or quantum state |1>. For example, considering a quantum system with five qubits, an exemplary quantum basis state of the five qubits is |0>|0>|1>|0>1>, represented as |00101> for short. The quantum state |00101> represents a configuration in which the first, second, and fourth qubits are in quantum state |0> and the third and fifth qubits are in quantum state |1>. For a quantum system with m qubits, 2 m There are m quantum basis states. Considering the principle of superposition, given two quantum states of a quantum system, a superposition of the quantum basis states is also a quantum state of the quantum system. For example, a superposition of the form a|00101>+b|11110>+c|11111> with complex numbers a, b and c is a quantum state of the quantum system. Mathematically, a quantum system of m qubits can be expressed as 2 m It can be represented as a dimensional vector space.

[0091] A qubit as described herein is to be understood as a physical qubit, i.e. a physical system that can be described by the theory of quantum mechanics in a two-dimensional vector space as described above. Examples of qubits are given below.

[0092] The quantum systems described herein may include or consist of multiple superconducting qubits, e.g., transmons or flux qubits. The superconducting qubits may include a first and a second superconducting loop. A superconducting current propagating clockwise and counterclockwise, respectively, in the first superconducting loop can form quantum basis states |1> and |0> of the superconducting qubit. Furthermore, a flux bias through the second superconducting loop can couple the quantum basis states |0> and |1>.

[0093] Alternatively, the quantum systems described herein may include systems of trapped ions, where the quantum basis states |0> and |1> of the qubit are formed by two levels of a Zeeman or hyperfine manifold, or across a forbidden optical transition of a positively charged ion, such as an alkaline earth or alkaline earth ion, such as Ca40+.

[0094] In yet another aspect, the quantum systems described herein may be realized using ultracold atoms, e.g. ultracold neutral alkali atoms, trapped in an optical lattice or large spacing lattice from a laser field. The atoms can be evolved towards a ground state using laser cooling. The quantum ground state of the qubit is formed by the ground state and higher Rydberg states of the atoms. The qubit can be addressed by laser light.

[0095] In yet another embodiment, the quantum systems described herein may be implemented using quantum dots. Quantum dot qubits may be fabricated from GaAs / AlGaAs heterostructures. The qubits are encoded with spin states, which can be prepared by adiabatically tuning the potential from a single well to a double well potential.

[0096] In yet another embodiment, the quantum systems described herein may be realized using impurities in solid crystals, such as NV centers, which are point defects in diamond crystals. Other impurities have been investigated, such as color centers associated with chromium impurities, rare earth ions in solid crystals, or defect centers in silicon carbide. The NV center has two unpaired electrons, providing a spin-1 ground state. This allows the identification of two long-lived sharp defect levels that can possibly be used to realize qubits together with the surrounding nuclear spins.

[0097] Quantum systems associated with neural networks

[0098] The neural networks described herein are associated with a quantum system. The quantum system includes a set of primary qubits and a set of auxiliary qubits. The quantum system is such that each node of the neural network, at least each node of the second set of nodes, at least each node of the first set of nodes, or at least each node of the first and second sets of nodes is associated with a primary qubit. Furthermore, each connection of the neural network may be associated with a pair of qubits consisting of a primary qubit and an auxiliary qubit.

[0099] An example of a quantum system is shown in Figure 5. The quantum system is associated with a neural network as shown in Figure 1b. The quantum system includes a set of primary qubits. The set of primary qubits may include primary qubits 5101, 5102, 5103 and 5104, primary qubits 5201, 5202 and 5203 and primary qubit 515. ij where i ranges from 1 to 4 and j ranges from 1 to 3. The quantum system may include an ancillary qubit 550. ij where i ranges from 1 to 4 and j ranges from 1 to 3.

[0100] Each node 10 in the first set of nodes (the "visible layer") shown in FIG. iis the primary qubit 510 shown in FIG. i can be associated with.

[0101] Each node 20 in the second set of nodes (the “hidden layer”) shown in FIG. j is the primary qubit 520 shown in FIG. j can be associated with.

[0102] Node 10 shown in Figure 1b i and node 20 j 15 for each connection between ij is the primary qubit 515 shown in FIG. ij and auxiliary qubit 550 ij The qubit pair can be associated with a pair of qubits consisting of:

[0103] The quantum system described herein comprises a set of primary qubits and a set of auxiliary qubits. The set of primary qubits may comprise at least three qubits, in particular at least eight qubits. The set of primary qubits may comprise K qubits, where K is 100 or more, for example 100-10000, and preferably 10000 or more. The set of auxiliary qubits may comprise at least three qubits, in particular at least eight qubits. The set of auxiliary qubits may comprise K' qubits, where K' is 100 or more, for example 100-10000, and preferably 10000 or more. It should be understood that the number of qubits shown in the figures described herein is shown for purposes of illustration and explanation, and the actual number of qubits may deviate therefrom.

[0104] The set of primary qubits may be arranged along at least a portion of a first two-dimensional lattice. For example, Figure 5 shows primary qubits arranged along a portion of a first two-dimensional lattice 580. Figure 7, described below, shows primary qubits arranged along a portion of a first two-dimensional lattice 780.

[0105] The set of ancillary qubits may be arranged along a portion of a second two-dimensional lattice. For example, Figure 5 shows ancillary qubits arranged along a portion of a second two-dimensional lattice 590. Figure 7, described below, shows ancillary qubits arranged along a portion of a second two-dimensional lattice 790.

[0106] A lattice, such as a two-dimensional lattice, may include multiple nodes or vertices spatially arranged along a regular grid. For example, in FIG. 5, the primary qubits, represented by black dots, correspond to the vertices of (a part of) a first two-dimensional lattice 580. The ancillary qubits, represented by squares, correspond to the vertices of (a part of) a second two-dimensional lattice 590.

[0107] A two-dimensional lattice is a planar structure and has the advantage that it offers a simpler spatial arrangement of qubits compared to, for example, a three-dimensional lattice or any irregular spatial arrangement.

[0108] A two-dimensional lattice can include multiple plaquette. A plaquette can be understood as a set of qubits, for example a set of four qubits that form a rectangle or a square. For example, a plaquette is composed of four qubits located at the four corners of the plaquette. For example, in FIG. 5, four primary qubits 5201, 5202, 515 12 and 515 11 form a plaquette of a first two-dimensional lattice 580. As another example, four ancillary qubits 550 11 , 550 12 , 550 22 and 550 21 form the plaquette of a second two-dimensional lattice 590.

[0109] A quantum system may include one or more ancillary qubits to "complete" the first and / or second two-dimensional lattices. For example, as shown in FIG. 5, the set of primary qubits may include ancillary qubit 585, which completes the upper left corner plaquette of first two-dimensional lattice 580. That is, the addition of ancillary qubit 585 completes the qubits 585, 5201, 515, 11 and 5101 form the plaquette.

[0110] According to some embodiments described herein, each node in the first set of nodes can be associated with a primary qubit, and each node in the second set of nodes can be associated with a primary qubit.

[0111] A primary qubit associated with a node of a neural network can be understood in the following sense: As mentioned above, the i-th node is a quantum computer with a classical spin s located at the i-th node. i Each spin s i For example, s i ∈{-1,1}. The association between the i-th node of the neural network and the primary qubit of the quantum system is given by the spin s located at the i-th node. i It may be necessary to provide a correspondence or encoding between the possible states of classical spin si and the possible quantum states of the qubit in question. In other words, each state of the classical spin si is encoded as a possible quantum state of the qubit, in particular a quantum basis state. Such encoding allows one to infer the state of the corresponding classical spin si when the qubit in question is in a particular quantum state (as revealed by measuring the qubit in question). For example, a possible encoding could be i =1 is considered to correspond to the quantum state |0>, and state s iThis includes assuming that =-1 corresponds to the quantum state |1>. Thus, if the qubit in question is in state |0>, then one can infer that the state of classical spin si is equal to 1. Similarly, if the qubit in question is in state |1>, then one can infer that the state of the associated classical spin si is -1.

[0112] In particular, all nodes in the first set of nodes (the visible layer) can be associated with a primary qubit. In this way, a training data configuration t=(t1,t2,t3,...) of a neural network can be encoded into a quantum state that includes primary qubits (i.e., "data qubits" as described herein) that correspond only to the visible layer. For example, a quantum state that encodes a training data configuration t=(t1,t2,t3,...) can have the form |τ1>|τ2>|τ3>..., where t i = 1, then |τ i >=|0>, and t i =-1, then |τ i >=|1>.

[0113] For example, referring to FIG. 5 in combination with FIG. 1b, primary qubit (data qubit) 510 i is node 10 i Node 10 i Each classical spin state located at i For example, the primary qubit 510 may be encoded as a corresponding quantum state of i The quantum states |0> and |1> of i t=(t1,t2,t3,t4) of nodes 101 to 104 can be encoded into quantum states of the form |τ1>|τ2>|τ3>|τ4> of qubits 5101 to 5104. This is shown in Figure 6. For example, if t=(1,1,1,-1) is an exemplary teacher data configuration, then the corresponding quantum state encoding this teacher data configuration is |0>|0>|0>|1>.

[0114] The neural network can be associated with a classical spin model as described herein. Each node of the first set of nodes can be associated with a primary qubit. The primary qubit can be configured to encode one or more states of the classical spin located at the node. The one or more states of the classical spin located at the node can be encoded as one or more quantum states of the primary qubit.

[0115] The present disclosure includes an association between the connections of a neural network and each primary qubit and ancillary qubit of a quantum system. The primary qubit associated with the connection between the i-th node and the j-th node of the neural network is represented by an interaction term s i s j In particular, each possible value of the interaction term may correspond to a possible quantum basis state of the primary qubit in question. For example, a value s i and s j Since is either 1 or -1, the interaction term s i s j (That is, the value s i and the value s j The product of | and | can also be either 1 or -1. In the encoding, one might have the value 1 correspond to the quantum state |0> and the value -1 correspond to the quantum state |1>. If the primary qubit in question is in state |0> (and thus a measurement on the qubit results in a 0), then the associated interaction term s i s j We can conjecture that is equal to 1. If the qubit in question is in state |1> (and thus the measurement results in 1), then we can conjecture that the relevant interaction term is -1.

[0116] For example, primary qubit 515 shown in FIG. ij If is in state |0> or |1>, the interaction term s i s j can be assigned the numerical value 1 or −1, respectively.

[0117] The neural network can be associated with the classical spin model described herein. Each connection of the neural network can be associated with a primary qubit. The connection can be between two nodes of the neural network. The primary qubit can be configured to encode a set of possible values ​​of the interaction between the classical spins located at the two nodes. The set of possible values ​​of the interaction between the classical spins located at the two nodes can be encoded as a set of quantum states of the primary qubit.

[0118] The ancillary qubit associated with the connection between the ith and jth nodes has the weight w of the connection in question. ij This may be understood in the sense that a correspondence or encoding is provided between the possible values ​​of w and the possible quantum states of the qubit in question, in particular the possible quantum basis states of the qubit. For example, ij Possible values ​​of can be 0 and 1. The value 0 can be considered to correspond to the quantum state |0>, and the value 1 can be considered to correspond to the quantum state |1>. If the qubit in question is measured and the measurement result is 0, then the associated weight w ij can be determined to be equal to 0. Similarly, if a measurement result is 1, the associated weight w ij can be determined to be 1.

[0119] For example, ancillary qubit 550 shown in FIG. ij If the connection is in state |0> or |1>, ij Weight of w ij can be assigned the numerical value 0 or 1, respectively.

[0120] Each connection of the neural network can be associated with an ancillary qubit. The ancillary qubit can be configured to encode a weight of the connection. A set of possible values ​​of the weight of the connection can be encoded as a set of quantum states of the ancillary qubit. Each connection of the neural network can be associated with both a primary qubit and an ancillary qubit.

[0121] The set of primary qubits may include qubits associated with a visible layer, qubits associated with connections of a neural network, and qubits associated with a hidden layer, as described below.

[0122] The set of primary qubits may include a subset of data primary qubits (or data qubits for short) consisting of primary qubits associated with the first set of nodes (visible layer). For example, in FIG. 5, the subset of data qubits may be understood as the set consisting of qubits 5101, 5102, 5103, and 5104. Each data qubit may be associated with a single node in the first set of nodes. Conversely, each node in the first set of nodes may be associated with a single data qubit. In other embodiments, there may be no such data qubits, and teacher data is input to the quantum system by two-body interactions as described herein.

[0123] The set of primary qubits may include a subset of connected primary qubits, which are primary qubits associated with connections in the neural network. For example, in FIG. 5, the connected primary qubits may be qubit 515. ij Each connected primary qubit can be associated with a single connection. Conversely, each connection can be associated with a single connected primary qubit. A subset of the data primary qubits may have no common elements with a subset of the connected primary qubits.

[0124] Each connection of the neural network can be associated with a pair of qubits, the pair consisting of a primary qubit and an ancillary qubit. The set of ancillary qubits can include a connected ancillary qubit subset consisting of the ancillary qubits associated with the connections of the neural network. For example, in FIG. 5, the connected ancillary qubit can be qubit 550. ij Each connected ancillary qubit can be associated with a single connection. Conversely, each connection can be associated with a single connected ancillary qubit. In some embodiments, each ancillary qubit can be a connected ancillary qubit.

[0125] The set of primary qubits may include a subset of the hidden primary qubits. Each node in the second set of nodes (hidden layer) may be associated with a hidden primary qubit. For example, in FIG. 5, the subset of hidden primary qubits may be understood as the set of qubits 5201, 5202, and 5203. Each hidden primary qubit may correspond to a single node in the second set of nodes. Conversely, each node in the second set of nodes may correspond to a single hidden primary qubit.

[0126] Similarly, each node in a further hidden layer (the third set of nodes, the fourth set of nodes described herein) can be associated with a hidden primary qubit.

[0127] The subset of data qubits, the subset of connected primary qubits, and the subset of hidden primary qubits may be disjoint sets (i.e., they may have no qubits in common).

[0128] Initialization

[0129] The methods described herein include initializing a quantum system to an initial quantum state. The initial quantum state may include a single qubit state or may be a (tensor) product of single qubit states. The initial quantum state may include a quantum state |α i >. The initial quantum state may have the form |α1>|α2>|α3>···. For example, the initial quantum state may have the form |+>|+>|+>···, where |+>=(|0>+|1>) / √2. It should be understood that the initial quantum state does not have to have the form |α1>|α2>|α3>···, i.e., the initial quantum state does not have to be a product of single qubit quantum states. The initial quantum state can be any separable or entangled state.

[0130] A quantum system can be initialized by cooling the quantum system (using a cooling device), by unitary evolution (e.g., using a quantum processing device described herein), by performing a measurement (using a measurement device described herein), etc., or by a combination thereof.

[0131] Each of the N rounds of operations may include preparing an initial quantum state for the round in question. That is, each round may include an initialization step. The initialization performed in a particular round may be performed before the sequence of unitary operations for the round is applied. The initial quantum state may be the same for all of the N rounds of operations. For example, the initial quantum state for each round may be a quantum state in which each qubit of a plurality of qubits is in state |+>=(|0>+|1>) / √2. Alternatively, different initial quantum states may be prepared for different rounds of operations.

[0132] Encoding training data in input qubits

[0133] The methods described herein include, for each teacher data configuration of a number of teacher data configurations obtained from a teacher data set, encoding the teacher data configuration with input qubits that form a subset of a set of primary qubits.

[0134] Each of the multiple training data configurations is, for example, based on the training data probability distribution P TD The number of teacher data configurations obtained from the teacher dataset may be one set sampled from the teacher dataset. The multiple teacher data configurations may include a first, a second, a third, a fourth, and a fifth teacher data configuration. The multiple teacher data configurations may include 10 or more, 100 or more, or 1000 or more teacher data configurations, each of which may be obtained by randomly sampling the teacher dataset described herein. The multiple teacher data configurations may include M teacher configurations, where M is greater than the size of the teacher dataset. For example, M may be 2 times, 10 times, or 10000 times the size of the teacher dataset. The multiple teacher data configurations may include multiple occurrences of the same teacher data configuration.

[0135] Training data probability distribution P TD By sampling the training dataset according to the side condition "P marg (t) ≒ P TD (t)" can be incorporated into the neural network, i.e., the resulting weights computed by the methods described herein satisfy the side conditions of the problem, thereby ensuring that the training of the neural network is performed in accordance with the desired side conditions.

[0136] The methods described herein include encoding supervised data configurations in input qubits that form a subset of the set of primary qubits. The supervised data configurations may be encoded in the input qubits by providing an interaction between each input qubit and an external entity (e.g., one or more fields, such as a magnetic field) or by providing an interaction between a group (e.g., a pair) of input qubits. The supervised data configurations may be encoded in the input qubits by biasing the input qubits toward a target quantum state of the input qubits.

[0137] In some embodiments, each node in the first set of nodes is associated with a primary qubit (a data qubit), such as, for example, qubits 5101, 5102, 5103, and 5104. The input qubits may be data qubits. Encoding the teacher data configuration in the input qubits includes biasing the data qubits towards a target quantum state, such that the target quantum state of the data qubits encodes the teacher data configuration in question.

[0138] As described herein, a training data configuration t = (t1, t2, t3,...) may be encoded into a target quantum state that may include data qubits. i Each number t i may be assigned to node i in the first set of nodes. i may correspond to or be encoded in the quantum basis state of the primary qubit associated with node i. For example, the number "1" may be encoded in the quantum basis state |0>, and the number "-1" may be encoded in the quantum basis state |1>. The target quantum state may include or be a product of single qubit quantum states. For example, for a teacher configuration t = (t1, t2, t3,...), the associated target quantum state may have the form |τ1>|τ2>|τ3>..., where t i = 1, then |τ i>=|0>, and t i =-1, then |τ i >=|1>. State |τ i > is the quantum state of the i-th data qubit.

[0139] Biasing a data qubit towards a target quantum state may be understood to mean that the primary qubit of interest is placed in a quantum state that is close to, i.e., approximates, the target quantum state, e.g., under the influence of one or more magnetic fields. For example, a strong magnetic field acting on each primary qubit of interest may achieve this. The more closely the quantum state approximates the target quantum state, the better the machine learning method will perform. However, some tolerance is acceptable, i.e., the quantum state need not be identical to the target quantum state.

[0140] The primary qubits, i.e., data qubits, associated with the first set of nodes can be biased towards a target quantum state by applying one or more fields to the data qubits. The one or more fields may include one or more magnetic fields, one or more electric fields, one or more electromagnetic fields, such as fields generated by lasers or microwaves, or a combination thereof. Additionally or alternatively, the bias may be obtained by applying one or more phase shifts resulting from mechanical deformation.

[0141] Data qubits can be biased towards a target quantum state by applying one or more two-body interactions to the data qubits. The two-body interactions can be formed from Pauli operators of the following form acting on pairs of data qubits:

number

[0142] At least a portion of the N operations may be performed while the primary qubit associated with the first set of nodes is biased towards the target state. In other words, the qubit in question may be biased towards the target state for the period during which at least a portion of the N operations are applied. This provides an improved approach to training neural networks. By performing iterative operations while biasing the quantum system towards a target state that encodes a teacher data configuration, the condition "P" is applied to the quantum system during the iterative procedure. marg (t) ≒ P TD (t)" can be applied. The iterative procedure minimizes the energy of the states of the quantum system under this side condition. Thus, the resulting weights calculated by the method described herein will satisfy the side condition of the problem. That is, it can be guaranteed that the training of the neural network is performed according to the desired side condition.

[0143] In other embodiments, the input qubit may be a primary qubit other than the primary qubits associated with the first and second sets of nodes. If there are no primary qubits, i.e., data qubits, associated with the first set of nodes, the number of qubits used in the method is reduced. The input qubit may be a connected primary qubit. For example, connected primary qubit 515 shown in FIG. 5 11 , 515 21 , 515 31 and 515 41may be input qubits. Primary qubits 5101, 5102, 5103 and 5104 may not be present. One or more interactions, for example two-body interactions, may be applied to the input qubits. The two-body interactions may be formed from Pauli operators σz(X)σz acting on the pair of input qubits. The two-body interactions may have an interaction strength represented by a parameter having a magnitude and a sign (referred to herein as the sign of the two-body interaction). The one or more two-body interactions may be configured to bias the pair of input qubits towards a respective target state that encodes the teacher data configuration t=(t1, t2, t3, . . .). For example, the two-body interactions may be configured to bias the pair of input qubits (515) shown in FIG. 5 towards a respective target state that encodes the teacher data configuration t=(t1, t2, t3, . . .). 11 ,515 21 ), (515 21 ,515 31 ), (515 31 ,515 41 ), or pair (515 1j ,515 2j ), (515 2j ,515 3j ), (515 3j ,515 4j ) (j = 1...3). The sign of each two-body interaction may depend on the parity of the corresponding pair of elements in the teacher data structure. For example, teacher data structure t = (t1, t2, t3, t4) has elements t1, t2, t3, t4, and the parity of two elements is their product (e.g., t1t2), which determines the sign of the two-body interaction. The two-body interaction is expressed as (t1t2)Cσ z 11 (X)σ z 21 ,···,(t3t4)Cσ z 31 (X)σ z 41 , or for any j = 1...3, (t1t2)Cσ z 1j (X)σ z 2j ,···,(t3t4)Cσ z 3j (X)σ z 4j where σz 11 Cubit 515 11 The two-body interaction may be applied over a period of time during which at least a portion of the N operations are applied.

[0144] Number of times

[0145] N operations can include 2 or more, specifically 10 or more, and more specifically 1000 or more operations.

[0146] A sequence of N unitary operations

[0147] The embodiments described herein include unitary operators or operations acting on a quantum system. A unitary operator is an operator (matrix) that acts on a quantum system. A unitary operator corresponds to a coherent time evolution of a quantum system. If a quantum system is initially in a state |Ψ>, applying a unitary operator U to the quantum system means that the state of the quantum system after execution of the unitary operator will be U|Ψ>.

[0148] The embodiments described herein include a Hamiltonian operator (referred to herein simply as a Hamiltonian) that acts on a quantum system. The Hamiltonian of a quantum system can represent one or more interactions of the quantum system. The eigenvalues ​​of the Hamiltonian correspond to the energy spectrum of the quantum system. The ground state of the Hamiltonian is the quantum state of the quantum system with the minimum energy. The ground state of the Hamiltonian can be thought of as the quantum state at zero temperature.

[0149] If H is the Hamiltonian of a quantum system, then the operator exp(itH) is a unitary operator, where t is a time parameter, i is the imaginary unit, and exp(·) is an exponential function. A unitary operator of the form exp(itH) is referred to herein as a unitary time evolution with respect to the Hamiltonian H. The evolution of a quantum system through a unitary time evolution exp(itH) of the Hamiltonian H can include turning on an interaction between multiple qubits of the quantum system, where the interaction is defined by the Hamiltonian H. The interaction can be switched on for a period t. The interaction can be switched off after the period t has elapsed.

[0150] There is likely to be at least a small amount of noise in the system. Unitary operators, such as the unitary time evolution of a Hamiltonian, do not need to be performed with 100% accuracy. It should be understood that the term "unitary operator" also includes near-unitary operators that are performed within a small margin of error, i.e., unitary operators that are performed in the presence of a small amount of noise.

[0151] As described herein, a sequence of unitary operators can be understood as an ordered list U1, U2,..., where each U i is a unitary operator. As described herein, each iteration of N operations involves evolving the quantum system by applying that iteration's sequence of unitary operators to the quantum system. Applying a sequence of unitary operators to a quantum system means that all unitary operators in the sequence are applied to the quantum system in the order prescribed by the sequence of unitary operators. Applying a sequence of unitary operators to a quantum system corresponds to evolving the quantum system by a total unitary operator that is the product of all unitary operators in the sequence of unitary operators. The products are performed in the order prescribed by the sequence.

[0152] For each of the N operations, the sequence of unitary operators may include two or more, specifically five or more, more specifically 200 or more unitary operators.

[0153] Short-distance unitary operators

[0154] Each unitary operator in a sequence of N respective unitary operators may be a short-distance unitary operator. The term "short-distance unitary operator" refers to a unitary operator that acts only on a subgroup of qubits, where any two qubits in the subgroup of qubits are separated from each other by a distance at most the interaction cutoff distance of the quantum system. A short-distance unitary operator does not act on qubits outside the subgroup of qubits. A subgroup of qubits may include one or more qubits. Mathematically, a short-distance unitary operator is at least approximately of the form A=A'(X)I, where (X) is a tensor product, A' is a unitary operator that acts on the subgroup of qubits, and I is an identity operator that acts on all qubits outside the subgroup of qubits.

[0155] The interaction cutoff distance described herein may be a certain distance. The interaction cutoff distance may be much smaller than the maximum qubit distance between qubits of a quantum system. For example, the interaction cutoff distance may be 30% or less, specifically 20% or less, more specifically 10% or less of the maximum qubit distance.

[0156] For a plurality of qubits arranged along a lattice, the short-range unitary operator may be an r-range unitary operator, where r is from 1 to 5, e.g., r = √2, 2, 3, 4 or 5. The r-range unitary operator may be a unitary operator that acts only on a subgroup of qubits of the plurality of qubits, where any pair of qubits in the subgroup of qubits is separated from each other by a distance of at most r times the fundamental distance of the lattice (the lattice constant).

[0157] 5, the fundamental distance or lattice constant of first two-dimensional lattice 580 may be given by the distance between two adjacent primary qubits of first two-dimensional lattice 580, e.g., the distance between qubits 5201 and 5202. Thus, for example, the distance between qubits 5201 and 5203 is twice the fundamental distance, and the distance between qubits 5201 and 5203 is twice the fundamental distance. 31 The distance between qubits 5201 and 515 is three times the fundamental distance. 12 The distance between qubits 5201 and 5203 is √2 times the fundamental distance. Therefore, the unitary operator acting only on qubits 5201 and 5203 is the r-range operator with r=2, and the distance between qubits 5201 and 515 is the fundamental distance between qubits 5202 and 515. 31 The unitary operator acting only on the qubits 5201 and 515 is the r-range operator with r=3. 12 A unitary operator that acts only on is the r-range operator with r=√2.

[0158] Regardless of the number of qubits in the quantum system, an operator that acts only on two qubits in the quantum system, where the two qubits are adjacent qubits ("nearest neighbor" qubits) of the quantum system, shall be considered to be a short-distance unitary operator. In the case of multiple qubits arranged on (a part of) a two-dimensional lattice, the distance between two adjacent qubits may be equal to the fundamental distance (lattice constant) of the two-dimensional lattice. For example, in FIG. 5, qubits 5201 and 5202 are two adjacent qubits of a first two-dimensional lattice 580. An operator that acts only on qubits 5201 and 5202 is an example of a short-distance operator.

[0159] Regardless of the number of qubits in a quantum system, simplicial unitary operators shall be considered to be short-range unitary operators. A simplicial unitary operator is an operator that acts on a single qubit of a quantum system, e.g., the i-th qubit. Mathematically, a simplicial unitary operator has the form A=A'(X)I, where A' is an operator that acts on a single qubit and I is the identity operator that acts on all qubits except for the single qubit.

[0160] Adaptation times

[0161] In some embodiments, the method includes feedforward of information, where the sequence of unitary operators applied in one operation may depend on a measurement result of measurements performed in one or more previous operations. The N operations may include one or more adaptive operations. For each adaptive operation, a unitary operator of the sequence of unitary operators of the adaptive operation is determined based on at least one measurement result of measurements performed in a previous one of the N operations.

[0162] The one or more adaptation rounds of operations may include a first adaptation round of operations. A method according to embodiments described herein may include processing, e.g., by a classical computing system as described herein, one or more measurements of an operation of an operation prior to the first adaptation round of operations. The method may include determining a sequence of unitary operators of the first adaptation round of operations in response to processing, e.g., by the classical computing system, one or more measurements of an operation of an operation prior to the first adaptation round of operations.

[0163] In some embodiments, the sequence of unitary operators applied in the adaptation round may depend on measurement results of two previous operations. The method may include, for example, processing, by a classical computing system, measurement results of two operations prior to the operation of the first adaptation round. The method may include determining a sequence of unitary operators for the operation of the first adaptation round as a function of, for example, processing, by a classical computing system, measurement results of two operations prior to the operation of the first adaptation round.

[0164] The one or more adaptation rounds of operation described herein may be multiple adaptation rounds, which may include two or more, and in some cases up to 10,000 or more, adaptation rounds.

[0165] measurement

[0166] According to an embodiment, each of the N operations includes performing a measurement of at least a portion of the quantum system. The measurements may be performed by a measurement device as described herein. For each of the N operations, the i-th measurement may be performed after applying the i-th sequence of unitary operators.

[0167] One or more qubits of a quantum system can be measured. One or more qubits may be measured individually. Performing a measurement of one or more qubits involves applying a Pauli operator to each of the one or more qubits, e.g., the Pauli operator σ z Performing a measurement of the one or more qubits may include performing a two outcome measurement for each of the one or more qubits. The two outcome measurements may provide one of two possible outcomes, e.g., 0 or 1.

[0168] The same measurement may be performed for at least two of the N operations. In other words, one or more quantum observables may be measured for a first of the N operations, and one or more quantum observables may be measured for a second of the N operations. Performing the same measurement for several operations allows the measurement results of the same measurement for different operations to be compared. For example, the energies of quantum states produced by different sequences of unitary operations may be compared to each other, as described herein.

[0169] For a quantum system including a plurality of K superconducting qubits, the qubit states |0> and |1> of the plurality of qubits can be measured with high fidelity using a measurement apparatus including a plurality of superconducting quantum interference devices, specifically, K hysteretic DC superconducting quantum interference devices and K RF superconducting quantum interference device latches controlled by bias lines, where the number of bias lines scales according to √K.

[0170] For quantum systems realised with trapped ions, measurements of the quantum system can be performed by fluorescence spectroscopy. There, when the ion is in one of two spin states, it undergoes a transition with a short lifetime. As a result, the ion in the driven state emits many photons, while the other ion remains dark. The emitted photons can be recorded by a CCD camera. Measurements in any direction on the Bloch sphere are performed by appropriate single-qubit pulses prior to fluorescence spectroscopy.

[0171] For quantum systems realized with cold atoms, qubits can be measured by selectively sweeping the ground state atoms and performing fluorescence imaging with single-site resolution.

[0172] In the case of quantum systems implemented with quantum dots, the qubits can be read out from a pulse sequence by rapid adiabatic passage.

[0173] Energy measurement comparison

[0174] The N operations may include the R1 operation. Applying the R1 sequence of unitary operators may result in a first quantum state of the quantum system. For example, |Ψ in 〉 may denote the initial quantum state of R1, and U1 may denote the unitary operator that is the product of all unitary operators in the sequence of unitary operators of R1. Thus, |Ψ1〉=U1|Ψ in > may be the first quantum state of a quantum system.

[0175] Methods according to embodiments described herein may include measuring the energy of the first quantum state, in particular the average energy or expected energy. The energy may be measured by a measurement device as described herein. Measuring the energy of the first quantum state may include measuring a Hamiltonian of the quantum system. Measuring the energy of the first quantum state may include measuring an expectation value of the Hamiltonian of the quantum system. The expectation value of the Hamiltonian H may be expressed as <Ψ1|H|Ψ1>. Measuring the energy of the first quantum state may include measuring a total Hamiltonian of the quantum system as described herein.

[0176] Measuring the energy involves measuring the qubits in the system individually, e.g., the Pauli operator σ z Based on the measurement results obtained from such measurements, the energy can be calculated, for example by a classical computing system as described herein.

[0177] The N operations can include R2 operations performed after R1. Applying the R2 sequence of unitary operators can result in a second quantum state of the quantum system. For example, |Ψ in 〉 is also the initial quantum state in time R2, i.e., the same initial state as in time R1, and U2 may denote the unitary operator that is the product of all unitary operators in the sequence of unitary operators in time R2. Therefore, |Ψ2〉=U2|Ψ in > may be a second quantum state of the quantum system.

[0178] Methods according to embodiments described herein may include measuring the energy of the second quantum state, in particular the average energy or expected energy. The energy may be measured by a measurement device as described herein. Measuring the energy of the second quantum state may include measuring a Hamiltonian of the quantum system. Measuring the energy of the second quantum state may include measuring an expectation value of the Hamiltonian of the quantum system. The measured Hamiltonian for the first quantum state may be the same as the measured Hamiltonian for the second quantum state. Measuring the energy of the second quantum state may include measuring a total Hamiltonian of the quantum system.

[0179] The method may include comparing, for example, by a classical computing system, an energy of the first quantum state to an energy of the second quantum state. The method may include determining, for example, by a classical computing system, a sequence of unitary operators to be applied to an R3th of the N operations, the R3th being performed after the R2th. The sequence of unitary operators to be applied to the R3th may be determined based on at least a comparison of the energy of the first quantum state to the energy of the second quantum state.

[0180] For example, if a comparison of the energy of the first quantum state with the energy of the second quantum state reveals that the energy of the first quantum state is less than that of the second quantum state, the operator or classical computing system may conclude that the first quantum state is closer to the minimum energy state of the measured Hamiltonian than the second quantum state. In view of this, the operator or classical computing system may reject the sequence of unitary operators in the R2th time and return to the sequence of unitary operations in the R1th time. Starting from the sequence of unitary operations in the R1th time, the operator or classical computing system may apply a small perturbation to the sequence by changing the parameters of the operators (e.g., the aforementioned parameters β, Ω, α or τ) by a small amount from the sequence. The resulting sequence may be a sequence of unitary operators applied in the R3th time.

[0181] Alternatively, if a comparison of the energy of the first quantum state with the energy of the second quantum state reveals that the energy of the first quantum state is greater than (or equal to) the energy of the second quantum state, the operator or classical computing system may conclude that the second quantum state is closer to the ground state of the measured Hamiltonian than the first quantum state. With this in mind, the operator or classical computing system may accept a sequence of unitary operators in the R2th time. Starting with a sequence of unitary operations in the R2th time, the operator or classical computing system may make small adjustments or perturbations to said sequence. The resulting adjusted sequence may be a sequence of unitary operators applied in the R3th time.

[0182] The method can proceed in a similar manner through all rounds of operations: (i) measure the energy of the quantum state obtained after applying the sequence of unitary operations of the current round, (ii) compare the measured energy of the current round with the measured energy of the previous round, (iii) reject the sequence of unitary operators of the current round and accept the sequence of unitary operations of the previous round if the measured energy of the current round is greater than the measured energy of the previous round, or accept the sequence of unitary operations of the current round if the measured energy of the current round is less than the measured energy of the previous round, and (iv) starting from the accepted sequence of unitary operations, perturb at least one parameter of the accepted unitary operators to obtain a sequence of operations for the next round of operations.

[0183] As the number of times N increases, a larger set of quantum states is prepared and the energy of the subsequent quantum states does not increase but gradually decreases. Thus, the energy gradually approaches the minimum energy of the measured Hamiltonian. As described herein, N operations can be performed while the teacher data configuration is encoded into the quantum system, for example by applying one or more fields that bias the data qubits towards a target state that encodes the teacher data configuration. In doing so, the quantum system is driven to a low-energy quantum state subject to the condition that the data qubits are biased towards the target state in question. This can be understood as a minimum energy search under side conditions that keep part of the system in a fixed state. The fixed part of the quantum system acts as an effective local field on the dynamic part. By repeating this process for an appropriate number of teacher data configurations, the approach can map the auxiliary qubits into quantum states that encode the solution to the machine learning problem, i.e., the initially unknown weights of the neural network upon measurement, while the condition P marg (t) ≒ P TD The objective of the present invention is to drive the trainer towards a quantum state that ensures that (t) is satisfied for each training data configuration t in the training data set.

[0184] Although an example of a method is described above in which the energy is measured over multiple adaptive rounds of operations, in alternative embodiments, the energy may be measured without performing any adaptive rounds of operations. For example, the method may include only non-adaptive rounds, where the energy of the Hamiltonian is measured over each round of operations. After all N rounds have been performed, the various measured energies may be compared to each other and the lowest energy may be selected.

[0185] Hamiltonian formalism

[0186] As described herein, the measurement of the quantum system may be an energy measurement. The measurement may be a measurement of the Hamiltonian of the quantum system. The Hamiltonian may have a particular form, as described below.

[0187] The measurements performed at each of the N operations may be measurements of a short-range Hamiltonian. A short-range Hamiltonian refers to a Hamiltonian that describes the interaction of multiple qubits, where no interaction occurs between qubits that are separated from each other by a distance that exceeds the interaction cutoff distance of the quantum system. For multiple qubits arranged along a lattice, the short-range Hamiltonian may be a Hamiltonian where no interaction occurs between qubits that are separated from each other by a distance that exceeds r times the fundamental distance (lattice constant) of the lattice. For example, r may be 1 to 5, e.g., r = √2, 2, 3, 4, or 5. Regardless of the number of qubits in the quantum system, the Plackett Hamiltonian is considered to be a short-range Hamiltonian.

[0188] The Hamiltonian measured in one operation may be a d-body Hamiltonian, where d is 2, 3, 4, 5, 6, 7, or 8. A d-body Hamiltonian may refer to a Hamiltonian that represents interactions of multiple qubits, where no coupling interactions occur between groups containing d+1 or more qubits, or at least such interactions are negligible. A d-body Hamiltonian may include interactions between groups containing d or fewer qubits. A d-body Hamiltonian may be the sum of multiple augend Hamiltonians, where each augend Hamiltonian represents coupling interactions between groups of d or fewer qubits. Each augend Hamiltonian may have the form A(X)I (at least approximately), where A is an operator acting on d or fewer qubits, and I is an identity operator acting on the rest of the quantum system.

[0189] The Hamiltonian, particularly the short-range Hamiltonian, measured in one operation may include a first sum including multiple first summand Hamiltonians, each of which may act on at least a primary qubit and an ancillary qubit.

[0190] Each first summand Hamiltonian may operate on a primary qubit and an ancillary qubit associated with the same connection of the neural network. For example, referring to FIG. 5, the first summand Hamiltonian may operate on primary qubit 515 ij and auxiliary qubit 550 ij , both of which are represented by nodes 510 of the neural network shown in FIG. 1b. i and 520 j It is associated with the connection between.

[0191] Each first augend Hamiltonian may be a two-body Hamiltonian. Each first augend Hamiltonian may have the form A(X)B, where A acts on a primary qubit and B acts on an ancillary qubit, e.g., the primary qubit and the ancillary qubit are associated with the same connection. For example, each augend Hamiltonian may be Jσ z(X)σ z where σ z is the Pauli operator and J is the coefficient.

[0192] The multiple first augend Hamiltonians may be configured such that each primary qubit is acted upon by at most one first augend Hamiltonian. In other words, a primary qubit may not be acted upon by any first augend Hamiltonian at all, or there may be a single first augend Hamiltonian acting upon the primary qubit in question. The multiple first augend Hamiltonians may be configured such that there are no two first augend Hamiltonians acting upon the same primary qubit.

[0193] The plurality of first augend Hamiltonians may be configured such that each ancillary qubit is acted upon by at most one, in particular exactly one, first augend Hamiltonian. The plurality of first augend Hamiltonians may be configured such that there are no two first augend Hamiltonians acting upon the same ancillary qubit.

[0194] The Hamiltonian, particularly the short-distance Hamiltonian, measured in one operation may include a second summation including multiple second augend Hamiltonians. Each second augend Hamiltonian may operate on multiple primary qubits, e.g., four primary qubits. Each second augend Hamiltonian may operate entirely within a set of primary qubits, i.e., each second augend Hamiltonian operates only on primary qubits. Each second augend Hamiltonian cannot operate on ancillary qubits. Each second augend Hamiltonian may have the form A(X)I, where A operates on one or more primary qubits and I is an identity operator that operates on a set of ancillary qubits.

[0195] The first-order qubits may be arranged along a portion of a two-dimensional lattice, for example as shown in Figures 5 and 7. Each second augend Hamiltonian may act along a plaquette of the two-dimensional lattice. For example, each second augend Hamiltonian may act on multiple first-order qubits located at the corners of a square. For example, each augend Hamiltonian may be Cσ z (X)σ z (X)σ z (X)σ z where σ z are the Pauli operators, and the four Pauli operators are located at the corners of the plaquette of the two-dimensional lattice. C is the coefficient.

[0196] The Hamiltonian measured in one operation may be a total Hamiltonian. The total Hamiltonian is equal to a first summation including a plurality of first summand Hamiltonians plus a second summation including a plurality of second summand Hamiltonians. For example, the total Hamiltonian may be a Hamiltonian of the form:

number

[0197] 1st and 2nd

[0198] Performing the N operations may include performing a plurality of first rounds of operations, each of which may include applying a sequence of unitary operators selected from a first family of unitary operators, and each of which may include performing a measurement of at least a portion of the quantum system.

[0199] Performing the N operations may include performing a plurality of second operations, each of which may include applying a sequence of unitary operators selected from a second family of unitary operators, the second family of unitary operators being different from the first family of unitary operators, and each of which may include performing a measurement of at least a portion of the quantum system.

[0200] The multiple second rounds of operations can be performed after or before the multiple first rounds of operations. There may be 2 or more, 10 or more, 100 or more, or 1000 or more first rounds. There may be 2 or more, 10 or more, 100 or more, or 1000 or more second rounds. The multiple first rounds may include one or more adaptation rounds as described herein. The multiple second rounds may include one or more adaptation rounds.

[0201] Thus, the N operations can be split into multiple first rounds and multiple second rounds, where the unitary operator applied in the first round is different from the unitary operator applied in the second round. An example is given below. Splitting the N rounds of appropriate first and second families into first and second rounds can improve the performance of the method, i.e., it can find a solution to the machine learning problem faster, compared to a method in which the N rounds all contain unitary operators taken from the same family.

[0202] For example, each unitary operator of the first family of unitary operators may operate entirely within the set of primary qubits. It will be understood that a unitary operator (or, equivalently, a Hamiltonian) operates entirely within the set of primary qubits means that the unitary operator does not operate on the ancillary qubits of the quantum system, or at least the effect of the unitary operation on the set of ancillary qubits is negligibly small, i.e., approximately zero. A unitary operation operating entirely within the set of primary qubits may have (approximately) the form U(X)I, where U is the unitary operation operating on the set of primary qubits and I is the identity operator operating on the set of ancillary qubits.

[0203] The set of primary qubits may include data qubits, connected primary qubits, and hidden primary qubits, as described herein. As described herein, each unitary operation applied to the quantum system in the first round of operations may not act on any data qubits, or may have a negligible effect on at least a subset of the data qubits. Each unitary operation applied in the first round may act on at least one or more connected primary qubits and / or one or more hidden primary qubits.

[0204] In some embodiments, each unitary operator of the first family of unitary operators may be a simplex unitary operator or a short-range unitary operator. Simplex unitary operators and / or short-range unitary operators may operate entirely within a set of primary qubits. A simplex unitary operator may be understood as an operator that operates on a single qubit of a quantum system.

[0205] In further embodiments, the primary qubits may be arranged along a portion of a first two-dimensional lattice as described herein. Each unitary operator of the first family of unitary operators may be a simplex unitary operator or a short-range unitary operator acting along a plaquette of the first two-dimensional lattice. The simplex unitary operator and / or the short-range unitary operator may act entirely within the set of primary qubits.

[0206] For example, a simplex unitary operator of the first family of operators may be an operator that acts on a single primary qubit, where the operator has the form:

number

[0207] For example, a short-range unitary operator acting along the plaquette of the first two-dimensional lattice may have the form

number

[0208] The sequence of unitary operators in each of the first operations may be an alternating sequence of the form A1B1A2B2.... k can be a product of two or more simplicial unitary operators. Each B k A can be a product of two or more short-range unitary operators, for example short-range operators acting along the plaquette of the first two-dimensional lattice. Each A k and B. k can operate entirely within the set of primary qubits. For example, each A k can be an operator of the form

number

number

[0209] The multiple first-round operations may include one or more adaptive first-round operations, and for each adaptive first-round operation, a unitary operator of the sequence of adaptive first-round unitary operators is determined based on at least one measurement result of a measurement performed in a previous first-round operation.

[0210] The first round may involve unitary operators acting only on the primary qubits, while the second round may involve at least some unitary operators acting on the ancillary qubits.

[0211] A second family of unitary operators may include one or more simple unitary operators acting on a single ancillary qubit, for example unitary operators of the form

number

[0212] The second family of unitary operators may include one or more unitary operators that operate jointly on primary qubits and ancillary qubits. In particular, at least some unitary operators of the second family may operate jointly on primary qubits and ancillary qubits that are associated with the same connection of the neural network. For example, with reference to FIG. 5, the unitary operators of the second family may operate jointly on primary qubits 515 and ancillary qubits 520 and 522. ij and auxiliary qubit 550 ij , both of which are represented by nodes 510 of the neural network shown in FIG. 1b. i and 520 jIt is associated with the connection between.

[0213] For example, the second family of unitary operators might include one or more operators of the form:

number

[0214] Each unitary operator in the second family of unitary operators can be a simplex unitary operator or a two-body unitary operator acting on a primary qubit and an ancillary qubit. A two-body unitary operator refers to a unitary operator that acts only on a subgroup of qubits that contains at most two qubits. A two-body unitary operator does not act on qubits outside the subgroup of qubits. Mathematically, a two-body unitary operator has the form U=U'(X)I (at least approximately), where U' is a unitary operator that acts on the subgroup of qubits and I is an identity operator that acts on all qubits outside the subgroup of qubits.

[0215] The sequence of unitary operators in each of the second operations may be an alternating sequence of the form C1D1C2D2.... Each C k can be a product of two or more simplicial unitary operators. Each D k can be a product of two or more short-range unitary operators, e.g., short-range two-body operators acting on a primary qubit and an ancillary qubit, respectively. For example, each C k may have the following form:

number

number

[0216] The multiple second-round operations may include one or more adaptive second-round operations, and for each adaptive second-round operation, a unitary operator of the sequence of adaptive second-round unitary operators is determined based on at least one measurement result of a measurement performed in a previous second-round operation.

[0217] Examples of deep networks and related quantum systems

[0218] 7 shows a further example of a quantum system associated with the deep neural network shown in FIG.

[0219] The quantum system shown in Figure 7 includes a set of primary qubits (black dots) arranged along a portion of a first two-dimensional lattice 780. The quantum system includes a set of ancillary qubits (squares) arranged along a portion of a second two-dimensional lattice 790. The first two-dimensional lattice has a plaquette. A plaquette is an elementary square of the first two-dimensional lattice with four primary qubits located at the corners of the elementary square. Similarly, a plaquette of the second two-dimensional lattice is an elementary square of the second two-dimensional lattice with four ancillary qubits located at the corners of the elementary square.

[0220] The quantum system shown in Figure 7 includes a region 710. Each node of the first set of nodes (visible layer) of Figure 2, i.e., node 210, is associated with a primary qubit 712 in region 710. In other words, primary qubit 712 is a data qubit. In some embodiments, as described herein, the data qubit may be omitted.

[0221] The quantum system shown in Figure 7 includes a region 720. Each connection between a first set of nodes (i.e., nodes 210) and a second set of nodes (i.e., nodes 220) in Figure 2 is associated with a primary qubit 722 and an ancillary qubit 724 in region 720.

[0222] The quantum system shown in Figure 7 includes a region 730. Each connection between the second set of nodes (i.e., node 220) and the third set of nodes (i.e., node 230) in Figure 2 is associated with a primary qubit 732 and an ancillary qubit 734 in region 730.

[0223] The quantum system shown in Figure 7 includes a region 740. Each connection between the third set of nodes (i.e., node 230) and the fourth set of nodes (i.e., node 240) in Figure 2 is associated with a primary qubit 742 and an ancillary qubit 744 in region 740.

[0224] The quantum system shown in Figure 7 includes a region 750. Each connection between the fourth set of nodes (i.e., node 240) and the fifth set of nodes (i.e., node 250) in Figure 2 is associated with a primary qubit 752 and an ancillary qubit 754 in region 750.

[0225] The qubits in regions 720, 730, 740 and 750 are connected qubits as described herein.

[0226] To better understand the layout of qubits in Figure 7, reference is made to Figure 8. Figure 8 shows the quantum system of Figure 7 again. The black dots represent the primary qubits in regions 710, 720, 730, 740 and 750. To avoid overloading the figure, ancillary qubits are not explicitly shown in Figure 8. Nevertheless, ancillary qubits can be provided as shown in Figure 7.

[0227] The qubits shown in Figure 8 are arranged along a portion of a grid (a two-dimensional lattice) having horizontal (left-right) and vertical (up-down) lines. As shown in Figure 8, when traversing the grid from top to bottom, the horizontal lines correspond to node 210 (a first set of nodes), node 230 (a third set of nodes), and node 250 (a fifth set of nodes), in that order. When traversing the grid from left to right, the vertical lines correspond to node 220 (a second set of nodes), and node 240 (a fourth set of nodes), in that order.

[0228] The group of horizontal lines corresponding to node 210 includes all primary qubits associated with node 210. This includes a) and b) below. a) Region 710 containing primary qubits (data qubits) associated with node 210. b) Region 720, which includes first-order qubits (connected qubits) associated with a connection between node 210 and node 220. In region 720, a first-order connected qubit is located at each intersection between a horizontal line corresponding to node 210 and a vertical line corresponding to node 220.

[0229] The group of horizontal lines corresponding to node 230 includes all primary qubits associated with node 230. This includes a) and b) below. a) A region 730 that includes first-order qubits (connected qubits) associated with a connection between node 220 and node 230. In region 730, a first-order connected qubit is located at each intersection between a horizontal line corresponding to node 230 and a vertical line corresponding to node 220. b) Region 740 contains primary qubits associated with connections between nodes 230 and 240. In region 740, a primary qubit is located at each intersection between a horizontal line corresponding to node 230 and a vertical line corresponding to node 240.

[0230] The group of horizontal lines corresponding to node 250 includes all of the first-order qubits associated with node 250. This includes region 750, which includes first-order qubits (connected qubits) associated with a connection between node 240 and node 250. In region 750, a first-order connected qubit is located at each intersection between the horizontal line corresponding to node 250 and the vertical line corresponding to node 240.

[0231] A quantum system may include multiple ancillary qubits to "complete" portions of the first and / or second two-dimensional lattice. For example, in FIG. 7, an additional 4×3 set of primary qubits (4 rows and 3 columns) may be placed in the area to the right of region 720 and above region 740 to complete the upper right corner of the first two-dimensional lattice 780. Similarly, an additional 4×3 set of ancillary qubits may be placed in the same area to complete the upper right corner of the second two-dimensional lattice 790. These ancillary qubits may not be operative when multiple operations and / or measurements are performed as described herein. Alternatively, these qubits may not be coupled to the remainder of the system.

[0232] The neural network described herein may be a deep neural network. As described herein, the neural network may include a third set of nodes and a plurality of connections between the second set of nodes and the third set of nodes. Each connection between a node of the second set of nodes and a node of the third set of nodes may have a weight. Each connection between a node of the second set of nodes and a node of the third set of nodes may be associated with both a primary qubit and an ancillary qubit. The ancillary qubit may be configured to encode the weight of the connection between the second set of nodes and the third set of nodes. After performing a plurality of operations as described herein, a measurement of the ancillary qubit may reveal the unknown weight of the problem.

[0233] The methods described herein may include measuring, e.g., by a measurement device, one or more ancillary qubits associated with one or more connections between the second set of nodes and the third set of nodes. The methods may include determining, e.g., by a classical computing system, a weight for at least one of the connections between the second set of nodes and the third set of nodes based on the measurement of the one or more ancillary qubits.

[0234] As described herein, the neural network may include a fourth set of nodes and a plurality of connections between the third set of nodes and the fourth set of nodes. Each connection between a node of the third set of nodes and a node of the fourth set of nodes may have a weight. Each connection between a node of the third set of nodes and a node of the fourth set of nodes may be associated with both a primary qubit and an ancillary qubit. The ancillary qubit may be configured to encode the weight of the connection between the third set of nodes and the fourth set of nodes. After performing multiple operations as described herein, a measurement of the ancillary qubit may reveal the unknown weight of the problem.

[0235] The methods described herein may include measuring, e.g., by a measurement device, one or more ancillary qubits associated with one or more connections between the third set of nodes and the fourth set of nodes. The methods may include determining, e.g., by a classical computing system, a weight for at least one of the connections between the third set of nodes and the fourth set of nodes based on the measurement of the one or more ancillary qubits.

[0236] The primary qubits may be arranged along at least a portion of a first two-dimensional lattice. The auxiliary qubits may be arranged along at least a portion of a second two-dimensional lattice. The quantum system may include a first region (e.g., region 710 of FIG. 7) including primary qubits, i.e., data qubits, associated with a first set of nodes. Instead of using data qubits, information in the teacher data configuration may be input in other ways, such as through two-body interactions as described herein. The quantum system may include a second region (i.e., region 720). The second region may be adjacent to the first region along a first direction (e.g., horizontally in FIG. 7). The second region may include primary qubits and auxiliary qubits associated with connections between the first set of nodes and the second set of nodes.

[0237] The neural network may be a deep neural network. The quantum system may include a third region (i.e., region 730). The third region may be adjacent to the second region along a second direction (e.g., vertical direction in FIG. 7). The second direction may be different from the first direction or may be perpendicular to the first direction. The third region may include a primary qubit and an auxiliary qubit associated with a connection between the second set of nodes and the third set of nodes. The quantum system may include a fourth region (i.e., region 740). The fourth region may be adjacent to the third region along a first direction (e.g., horizontal direction in FIG. 7). The fourth region may include a primary qubit and an auxiliary qubit associated with a connection between the third set of nodes and the fourth set of nodes. The quantum system may include a fifth region (i.e., region 750). The fifth region may be adjacent to the fourth region along the second direction. The fifth region can include a primary qubit and an ancillary qubit associated with a connection between the fourth set of nodes and the fifth set of nodes of the neural network.

[0238] According to a further embodiment, there is provided a method of machine learning using a quantum system. The method includes providing a quantum system including a set of primary qubits and a set of auxiliary qubits. A neural network is associated with the quantum system. The neural network includes a first set of nodes, a second set of nodes, and a plurality of connections between the first set of nodes and the second set of nodes, each connection having a weight. The neural network includes a teacher data set including a plurality of configurations of the first set of nodes. Each node of the first set of nodes can be associated with a primary qubit. Each connection of the plurality of connections is associated with both a primary qubit and an auxiliary qubit. The method further includes initializing the quantum system to an initial quantum state. The method further includes, for each configuration in the set sampled from the teacher data set, the following steps: Encoding a configuration with input qubits forming a subset of the set of primary qubits, such as by applying one or more fields to bias primary qubits associated with a first set of nodes toward a target state, where the target state encodes the configuration. performing a plurality of first rounds of operations, where each first round includes (a1) applying a sequence of unitary operators selected from a first family of unitary operators, and (a2) performing a measurement of at least a portion of the quantum system. performing a plurality of second rounds of operations, where each second round includes (b1) applying a sequence of unitary operators selected from a second family of unitary operators, where the second family of unitary operators is different from the first family of unitary operators, and (b2) performing a measurement of at least a portion of the quantum system. The method further includes performing measurements of the one or more ancillary qubits and determining a weight for at least one of the connections of the plurality of connections based on the measurements of the at least one or more ancillary qubits.

[0239] According to a further embodiment, there is provided an apparatus for performing machine learning using a quantum system. The apparatus includes a quantum system including a set of primary qubits and a set of ancillary qubits as described herein. A neural network is associated with the quantum system. The neural network includes a first set of nodes, a second set of nodes, and a plurality of connections between the first set of nodes and the second set of nodes, each connection having a weight as described herein. The neural network includes a teacher data set including a plurality of configurations of the first set of nodes as described herein. Each connection of the plurality of connections is associated with both a primary qubit and an ancillary qubit. The apparatus includes a quantum processing device for evolving the quantum system according to one or more short-range unitary operators. The apparatus includes a coding device for interacting with input qubits forming a subset of the set of primary qubits. The coding device may be configured to apply a bias to the primary qubits associated with the first set of nodes. The apparatus includes a measurement device for measuring at least a portion of the quantum system. The apparatus is configured to initialize the quantum system to an initial quantum state. The apparatus is configured to perform the following steps for each configuration of the number of configurations obtained from the teacher data set: encoding, by an encoding device, a configuration in the input qubits by interacting with the input qubits, e.g., by biasing primary qubits associated with the first set of nodes toward a target state, where the target state encodes the configuration. performing N operations, each of which includes applying, by a quantum processing device, a sequence of unitary operators acting on at least a portion of a quantum system including the input qubit, and performing, by a measurement device, a measurement of at least a portion of the quantum system. The apparatus is configured to perform measurements of one or more ancillary qubits with a measurement device. The apparatus is configured to determine a weight of at least one of the connections of the plurality of connections based on the measurements of the at least one or more ancillary qubits. The at least one weight can be determined by a classical computation system included in the apparatus.

[0240] 9 illustrates an apparatus 900 according to embodiments described herein. The apparatus includes a quantum system 910 as described herein. The apparatus includes a quantum processing device 920 for evolving the quantum system 910 according to one or more short-range unitary operators as described herein. The apparatus 900 includes an encoding device 930 for encoding a configuration of input qubits forming a subset of the set of primary qubits, e.g., by applying biases to primary qubits associated with a first set of nodes, as described herein or otherwise. The apparatus 900 includes a measurement device 940 for measuring at least a portion of the quantum system 910, as described herein.

[0241] The apparatus 900 described herein may include a classical computing system 950. The classical computing system 950 may be coupled to at least one of the quantum processing device 920, the encoding device 930, and the measurement device 940. The classical computing system 950 may be configured to determine a weight for at least one of the connections of the neural network based at least on a measurement of one or more ancillary qubits performed by the measurement device 940.

[0242] A classical computing system may refer to a computing system that operates using classical bits. A classical computing system may include a central processing unit (CPU) for processing information using classical bits and / or a memory for storing information using classical bits. A classical computing system may include one or more classical computers, such as a personal computer (PC), and / or a network of classical computers.

[0243] The apparatus described herein may include a controller. The controller may include or be a classical computing system described herein. The controller may be coupled to the quantum processing device, the coding device, and / or the measurement device. The controller may be configured to select, e.g., randomly select, configurations from a teacher data set. The controller may be configured to perform at least one of the following steps for each configuration in the teacher data set: Encoding a configuration of input qubits forming a subset of the set of primary qubits, e.g., instructing an encoding device to bias primary qubits associated with a first set of nodes towards a target state, where the target state encodes the configuration. For each of the N times, instructing the quantum processing device to apply a sequence of unitary operators to the quantum system for the time in question, and instructing the measurement device to perform a measurement of at least a portion of the quantum system.

[0244] The controller may be configured to instruct the measurement device to perform a measurement of the one or more ancillary qubits. The controller may be configured to determine a weight for at least one of the connections of the plurality of connections based on the measurement of the at least one or more ancillary qubits.

[0245] The encoding device may include a field generating device for applying one or more fields, for example a magnetic field, an electric field or a laser field, to at least a portion of the quantum system.

[0246] The apparatus described herein may be configured to perform the method according to any of the embodiments described herein.

[0247] While the foregoing is directed to several embodiments of the present invention, other and further embodiments may be devised without departing from the scope thereof, as determined by the claims.

Claims

1. A machine learning method using a quantum system, comprising: The method includes providing a quantum system comprising a set of primary qubits and a set of ancillary qubits, the quantum system having associated therewith a neural network, the neural network comprising a first set of nodes (10 １ , 10 ２ , 10 ３ , 10 ４ ; 210) and a second set of nodes (20 １ , 20 ２ , 20 ３ 220), and a plurality of connections (15) between the first set of nodes and the second set of nodes, each connection having a weight. ｉｊ 215), and a training data set comprising a plurality of configurations for the first set of nodes, wherein each connection (15) of the plurality of connections ｉｊ ;215) is the primary qubit (515 ｉｊ ; 722) and an ancillary qubit (550 ｉｊ 724), wherein the ancillary qubits are configured to encode the weights of the connections; The method further comprises the steps of initializing the quantum system to an initial quantum state; and, for each configuration of a plurality of configurations obtained from the training data set, encoding the configuration in input qubits forming a subset of the set of primary qubits; and performing N operations, each operation comprising applying a sequence of unitary operators acting on at least a portion of the quantum system and performing a measurement of at least a portion of the quantum system. The method further comprises performing measurements of one or more ancillary qubits; and determining a weight for at least one connection of the plurality of connections based at least on the measurements of the one or more ancillary qubits. Machine learning methods.

2. The first set of nodes (10 １ , 10 ２ , 10 ３ , 10 ４ Each node of the １ , 510 ２ , 510 ３ , 510 ４ 712), and wherein encoding a configuration on the input qubits includes biasing the primary qubits associated with a first set of nodes toward a target state, the target state encoding the configuration; the primary qubit associated with the first set of nodes is biased towards the target state by applying one or more fields to at least a portion of the quantum system. The machine learning method of claim 1 .

3. The N operations include one or more adaptation rounds of operations, and for each adaptation round of operations, a unitary operator of the sequence of unitary operators of the adaptation round is determined based on at least one measurement result of a measurement performed in a previous round of the N operations. The machine learning method according to claim 1 or 2.

4. each unitary operator in the sequence of unitary operators of each of the N operations is a short-distance unitary operator; The machine learning method according to any one of claims 1 to 3.

5. the measurement performed in each of the N operations is a measurement of a short-range Hamiltonian; The machine learning method according to any one of claims 1 to 4.

6. the short-distance Hamiltonian has a first sum including a plurality of first summand Hamiltonians, each first summand Hamiltonian acting on at least a primary qubit and an ancillary qubit; The machine learning method according to claim 5.

7. the short-distance Hamiltonian has a second summation including a plurality of second summand Hamiltonians, each second summand Hamiltonian operating entirely within a set of primary qubits; the primary qubits are arranged along at least a portion of a two-dimensional lattice (580, 780), and each second summand Hamiltonian acts along a plaquette of the two-dimensional lattice; The machine learning method according to claim 5 or 6.

8. The step of performing N operations includes: performing a plurality of first round operations, each first round comprising: (a1) applying a sequence of unitary operators selected from a first family of unitary operators to a portion of the quantum system, where at least some of the unitary operators in the first family of unitary operators act on the input qubits; and (a2) performing a measurement of at least a portion of the quantum system; performing a plurality of second round operations, each second round comprising: (b1) applying a sequence of unitary operators selected from a second family of unitary operators to a portion of the quantum system, the second family of unitary operators being different from the first family of unitary operators, and at least some unitary operators of the second family of unitary operators acting on the ancillary qubits; and (b2) performing a measurement of at least a portion of the quantum system; each unitary operator of the first family of unitary operators acts entirely within the set of primary qubits; The machine learning method according to any one of claims 1 to 7.

9. at least some of the unitary operators of the second family of unitary operators act on a primary qubit and an ancillary qubit together; The machine learning method of claim 8.

10. each unitary operator of the second family of unitary operators is a simplex unitary operator or a two-body unitary operator acting on a primary qubit and an ancillary qubit; The machine learning method according to claim 8 or 9.

11. the primary qubits are arranged along at least a portion of a first two-dimensional lattice (580, 780) and / or the ancillary qubits are arranged along at least a portion of a second two-dimensional lattice (590, 790); The machine learning method according to any one of claims 1 to 10.

12. A machine learning method using a quantum system, comprising: The method includes providing a quantum system comprising a set of primary qubits and a set of ancillary qubits, the quantum system having associated therewith a neural network, the neural network comprising a first set of nodes (10 １ , 10 ２ , 10 ３ , 10 ４ ; 210) and a second set of nodes (20 １ , 20 ２ , 20 ３ 220), and a plurality of connections (15) between the first set of nodes and the second set of nodes, each connection having a weight. ｉｊ 215), and a training data set comprising a plurality of configurations for the first set of nodes, wherein each connection (15) of the plurality of connections ｉｊ ;215) is the primary qubit (515 ｉｊ ; 722) and an ancillary qubit (550 ｉｊ 724), The method further comprises initializing the quantum system to an initial quantum state; for each configuration in a set sampled from the teacher data set, encoding a configuration in input qubits forming a subset of the set of primary qubits; performing a plurality of first round operations, each first round comprising: (a1) applying a sequence of unitary operators selected from a first family of unitary operators; and (a2) performing a measurement of at least a portion of the quantum system; and performing a plurality of second round operations, each second round comprising: (b1) applying a sequence of unitary operators selected from a second family of unitary operators, the second family of unitary operators being different from the first family of unitary operators; and (b2) performing a measurement of at least a portion of the quantum system. The method further comprises performing measurements of one or more ancillary qubits; and determining a weight for at least one connection of the plurality of connections based at least on the measurements of the one or more ancillary qubits. Machine learning methods.

13. An apparatus (900) for performing machine learning using a quantum system, comprising: The apparatus comprises: A quantum system (910) comprising a set of primary qubits and a set of ancillary qubits, the quantum system having associated therewith a neural network having a first set of nodes, a second set of nodes, a plurality of connections between the first set of nodes and the second set of nodes, each connection having a weight, and a teacher data set comprising a plurality of configurations for the first set of nodes, each connection of the plurality of connections being associated with both a primary qubit and an ancillary qubit, the ancillary qubit being configured to encode the weight of the connection; a quantum processing device (920) for evolving said quantum system according to one or more short-range unitary operators; an encoding device (930) for interacting with input qubits forming a subset of the set of primary qubits; a measurement device (940) for measuring at least a portion of the quantum system; the apparatus is configured to perform the steps of initializing the quantum system to an initial quantum state; for each configuration of a plurality of configurations obtained from the teacher data set, encoding the configuration on the input qubits by interacting with the input qubits with the encoding device; and performing N operations, each operation comprising: applying, by the quantum processing device, a sequence of unitary operators acting on at least a portion of the quantum system; and performing, by the measurement device, a measurement of at least a portion of the quantum system; the apparatus is further configured to perform, with the measurement device, measurements of one or more ancillary qubits; and determining a weight for at least one connection of the plurality of connections based at least on the measurements of the one or more ancillary qubits. Machine learning execution device.

14. the primary qubits are arranged along at least a portion of a first two-dimensional lattice and / or the ancillary qubits are arranged along at least a portion of a second two-dimensional lattice; the encoding device comprising a field generating device for applying one or more fields to at least a portion of the quantum system; At least one of the following applies: The machine learning execution device of claim 13.

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