Optimizing High-Efficiency Quantum Memory with Quantum Machine Learning for Near-Term Quantum Devices.

Quantum memories are a fundamental of any global-scale quantum Internet, high-performance quantum networking and near-term quantum computers. A main problem of quantum memories is the low retrieval efficiency of the quantum systems from the quantum registers of the quantum memory. Here, we define a novel quantum memory called high-retrieval-efficiency (HRE) quantum memory for near-term quantum devices. An HRE quantum memory unit integrates local unitary operations on its hardware level for the optimization of the readout procedure and utilizes the advanced techniques of quantum machine learning. We define the integrated unitary operations of an HRE quantum memory, prove the learning procedure, and evaluate the achievable output signal-to-noise ratio values. We prove that the local unitaries of an HRE quantum memory achieve the optimization of the readout procedure in an unsupervised manner without the use of any labeled data or training sequences. We show that the readout procedure of an HRE quantum memory is realized in a completely blind manner without any information about the input quantum system or about the unknown quantum operation of the quantum register. We evaluate the retrieval efficiency of an HRE quantum memory and the output SNR (signal-to-noise ratio). The results are particularly convenient for gate-model quantum computers and the near-term quantum devices of the quantum Internet.

Introduction

Quantum memories are a fundamental of any global-scale quantum Internet[1-6]. However, while quantum repeaters can be realized without the necessity of quantum memories[1,3], these units, in fact, are required for guaranteeing an optimal performance in any high-performance quantum networking scenario[3,4,7-32]. Therefore, the utilization of quantum memories still represents a fundamental problem in the quantum Internet[33-42], since the near-term quantum devices (such as quantum repeaters[5,6,8,32,43-47]) and gate-model quantum computers[48-59] have to store the quantum states in their local quantum memories[43-47,60-84]. The main problem here is the efficient readout of the stored quantum systems and the low retrieval efficiency of these systems from the quantum registers of the quantum memory. Currently, no general solution to this problem is available, since the quantum register evolves the stored quantum systems via an unknown operation, and the input quantum system is also unknown, in a general scenario[4,5,7-9,11,12]. The optimization of the readout procedure is therefore a hard and complex problem. Several physical implementations have been developed in the last few years[85-105]. However, these experimental realizations have several drawbacks, in general because the output signal-to-noise ratio (SNR) values are still not satisfactory for the construction of a powerful, global-scale quantum communication network. As another important application field in quantum communication, the methods of quantum secure direct communication[106-109] also require quantum memory.

Here, we define a novel quantum memory called high-retrieval-efficiency (HRE) quantum memory for near-term quantum devices. An HRE quantum memory unit integrates local unitary operations on its hardware level for the optimization of the readout procedure. An HRE quantum memory unit utilizes the advanced techniques of quantum machine learning to achieve a significant improvement in the retrieval efficiency[110-112]. We define the integrated unitary operations of an HRE quantum memory, prove the learning procedure, and evaluate the achievable output SNR values. The local unitaries of an HRE quantum memory achieve the optimization of the readout procedure in an unsupervised manner without the use of any labeled data or any training sequences. The readout procedure of an HRE quantum memory is realized in a completely blind manner. It requires no information about the input quantum system or about the quantum operation of the quantum register. (It is motivated by the fact that this information is not accessible in any practical setting).

The proposed model assumes that the main challenge is the recovery the stored quantum systems from the quantum register of the quantum memory unit, such that both the input quantum system and the transformation of the quantum memory are unknown. The optimization problem of the readout process also integrates the efficiency of the write-in procedure. In the proposed model, the noise and uncertainty added by the write-in procedure are included in the unknown transformation of the QR quantum register of the quantum memory that results in a σ mixed quantum system in QR.

The novel contributions of our manuscript are as follows:

We define a novel quantum memory called high-retrieval-efficiency (HRE) quantum memory.

An HRE quantum memory unit integrates local unitary operations on its hardware level for the optimization of the readout procedure and utilizes the advanced techniques of quantum machine learning.

We define the integrated unitary operations of an HRE quantum memory, prove the learning procedure, and evaluate the achievable output signal-to-noise ratio values. We prove that local unitaries of an HRE quantum memory achieve the optimization of the readout procedure in an unsupervised manner without the use of any labeled data or training sequences.

We evaluate the retrieval efficiency of an HRE quantum memory and the output SNR.

The proposed results are convenient for gate-model quantum computers and near-term quantum devices.

This paper is organized as follows. Section 2 defines the system model and the problem statement. Section 3 evaluates the integrated local unitary operations of an HRE quantum memory. Section 4 proposes the retrieval efficiency in terms of the achievable output SNR values. Finally, Section 5 concludes the results. Supplemental material is included in the Appendix.

System Model and Problem Statement

System model

Let be an unknown input quantum system formulated by n unknown density matrices,where , and .

The input system is received and stored in the QR quantum register of the HRE quantum memory unit. The quantum systems are d-dimensional systems ( for a qubit system). For simplicity, we focus on dimensional quantum systems throughout the derivations.

The U unknown evolution operator of the QR quantum register defines a mixed state σ aswhere , .

Let us allow to rewrite (2) for a particular time t, , where T is a total evolution time, via a mixed system , aswhere is an unknown evolution matrix of the QR quantum register at a given t, with a dimensionwith , , while is an unknown complex quantity, defined asand

Then, let us rewrite from (3) aswhere is as in (1), and is an unknown residual density matrix at a given t.

Therefore, (7) can be expressed as a sum of M source quantum systems,where is the m-th source quantum system and , wherein our setting, sinceand

In terms of the M subsystems, (3) can be rewritten aswhere is a complex quantity associated with an m-th source system,with , , and

The aim is to find the V inverse matrix of the unknown evolution matrix U in (2), asthat yields the separated readout quantum system of the HRE quantum memory unit for , such that for a given t,where

For a total evolution time T, the target σ density matrix is yielded at the output of the HRE quantum memory unit, aswith a sufficiently high SNR value,where x is an SNR value that depends on the actual physical layer attributes of the experimental implementation.

The problem is therefore that both the input quantum system (1) and the transformation matrix U in (2) of the quantum register are unknown. As we prove, by integrating local unitaries to the HRE quantum memory unit, the unknown evolution matrix of the quantum register can be inverted, which allows us to retrieve the quantum systems of the quantum register. The retrieval efficiency will be also defined in a rigorous manner.

Problem statement

The problem statement is as follows.

Let M be the number of source systems in the QR quantum register such that the sum of the M source systems identifies the mixed state of the quantum register. Let m be the index of the source system, , such that identifies the unknown input quantum system stored in the quantum register (target source system), while are some unknown residual quantum systems. The input quantum system, the residual systems, and the transformation operation of the quantum register are unknown. The aim is then to define local unitary operations to be integrated on the HRE quantum memory unit for an HRE readout procedure in an unsupervised manner with unlabeled data.

The problems to be solved are summarized in Problems 1–4.

Problem 1.

Find an unsupervised quantum machine learning method, U, for the factorization of the unknown mixed quantum system of the quantum register via a blind separation of the unlabeled quantum register. Decompose the unknown mixed system state into a basis unitary and a residual quantum system.

Problem 2.

Define a unitary operation for partitioning the bases with respect to the source systems of the quantum register.

Problem 3.

Define a unitary operation for the recovery of the target source system.

Problem 4.

Evaluate the retrieval efficiency of the HRE quantum memory in terms of the achievable SNR.

The resolutions of the problems are proposed in Theorems 1–4.

The schematic model of an HRE quantum memory unit is depicted in Fig. 1.

Figure 1

The schematic model of a high-retrieval-efficiency (HRE) quantum memory unit. The HRE quantum memory unit contains a QR quantum register and integrated local unitary operations. The input quantum systems, , are received and stored in the quantum register. The state of the QR quantum register defines a mixed state, , where . The stored density matrices of the quantum register are first transformed by a , a quantum machine learning unitary (depicted by the orange-shaded box) that implements an unsupervised learning for a blind separation of the unlabeled input, and decomposable as , where is a factorization unitary, is the quantum constant transform with a windowing function for the localization of the wave functions of the quantum register, is a basis partitioning unitary, while is the inverse of . The result of is processed further by the unitary (depicted by the green-shaded box) that realizes the inverse quantum discrete short-time Fourier transform (DSTFT) operation (depicted by the yellow-shaded box), and by the (quantum discrete Fourier transform) unitary to yield the desired output .

The procedures realized by the integrated unitary operations of the HRE quantum memory are depicted in Fig. 2.

Figure 2

Detailed procedures of an HRE quantum memory. The unknown input quantum system is stored in the quantum register that realizes an unknown transformation. The density matrix of the quantum register is the sum of source systems, where source system identifies the valuable unknown input quantum system stored in the quantum register, while identifies an unknown undesired residual quantum system. The unitary evaluates bases for the source system and defines a auxiliary quantum system. The unitary is a preliminary operation for the partitioning of the bases onto clusters via unitary . The unitary regroups the bases with respect to the source systems. The results are then processed by the and unitaries to extract the source system on the output of the memory unit.

Experimental implementation

An experimental implementation of an HRE quantum memory in a near-term quantum device[52] can integrate standard photonics devices, optical cavities and other fundamental physical devices. The quantum operations can be realized via the framework of gate-model quantum computations of near-term quantum devices[52-56], such as superconducting units[53]. The application of a HRE quantum memory in a quantum Internet setting[1,2,4-6] can be implemented via noisy quantum links between the quantum repeaters[8,32,43-47] (e.g., optical fibers[7,62,113], wireless quantum channels[27,28], free-space optical channels[114]) and fundamental quantum transmission protocols[24,115-117].

Integrated Local Unitaries

This section defines the local unitary operations integrated on an HRE quantum memory unit.

Quantum machine learning unitary

The U quantum machine learning unitary implements an unsupervised learning for a blind separation of the unlabeled quantum register. The U unitary is defined aswhere U is a factorization unitary, U is the quantum constant Q transform, U is a partitioning unitary, while is the inverse of .

Factorization unitary

Theorem 1.

(Factorization of the unknown mixed quantum system of the quantum register). The U unitary factorizes the unknown mixed quantum system of the QR quantum register into a unitary , with a Hamiltonian and application time , and into a system , where , , and , and where T is the evolution time, M is the number of source systems of , and K is the number of bases.

Proof. The aim of the U factorization unitary is to factorize the mixed quantum register (2) into a basis matrix U and a quantum system , aswhere U is a complex basis matrix, defined asand is a complex matrix, defined aswherewhere , and , while is the total number of bases of , while is a complex quantity, as

The first part of the problem is therefore to find (22), where is a unitary that sets a computational basis for in (25), defined aswhere H is a Hamiltonian, aswhere G is the eigenvalue of basis , , while is the application time of u.

The second part of the problem is to determine W, aswhere is a system state, that formulates aswhere is an approximation of ,where is defined in (14).

As follows, for the total evolution time T, can be defined asand the challenge is to evaluate (31) as a decomposition

Thus, by applying of the u unitaries for the total evolution time T, is aswhere K is the number of bases associated with the m-th source system,and , .

In our setting , and our aim is to get the system state on the output of the HRE quantum memory, thus a target output system state is defined aswhere K1 is the number of bases for source system , .

Let rewrite the system state (32) asand letand

Then, let be a density matrix associated with , defined asand letbe the density matrix associated with (36).

The aim of the estimation is to minimize the quantum relative entropy function taken between and , thus an objective function for is defined via (37) and (38) as

To achieve the objective function in (41), a factorization method is defined for that is based on the fundamentals of Bayesian nonnegative matrix factorization[118-127] (Footnote: The factorization unitary applied on the mixed state of the quantum register is analogous to a Poisson-Exponential Bayesian nonnegative matrix factorization[118-121] process). The method adopts the Poisson distribution as likelihood function and the exponential distribution for the control parameters[118-121] and defined for the controlling of and .

Let and from (29) be defined via the control parameters and as exponential distributionswith mean , andwith mean .

Using (41), (42) and (43), a log likelihood functioncan be defined asthus the objective function can be rewritten via as (45)

The problem is therefore can be reduced to determine the model parametersthat are treated as latent variables for the estimation of the control parameters[118-121,125-127]

A maximum likelihood estimation of (47) is aswhere is some distribution, that identifies an incomplete estimation problem.

The estimation of (47) can also be yielded from a maximization of a marginal likelihood function aswhere is a complex matrix, ,wherewithwhere

The quantity in (54) can be estimated via (42) and (43) as

Using (54), in (29) can be rewritten as

However, since the exact solution does not exists[118-121], since it would require the factorization of , such that are unknown.

This problem can be solved by a variational Bayesian inference procedure[118-121,125-127], via the maximization of the lower bound of a likelihood function where is a variational distribution, while is the entropy of variational distribution ,and where is a joint variational distribution, asfrom which distribution can be approximated as[118-121]

The function in (57) is related to (50) as

The result in (59) therefore also determines the number of bases selected for the factorization unitary . The variational distributions , and are determined for the unitary U as follows.

Let refer to the variational distribution of a given ,

Since only the joint (posterior) distribution is obtainable, the variational distributions have to be evaluated aswhere is the expectation function of the variational distribution of i, such that , where is as in (62), withfor some functions and , andfor some constant b, (note: for simplicity, we use for the expectation function), whilewhere is the Dirac delta function, while is the Gamma function,

By utilizing a variational Poisson–Exponential Bayesian learning[118-121], these variational distributions can be evaluated as follows.

The variational distribution is aswhere is a multinomial distribution, while is a multinomial parameterwhile the variational distribution is aswhere is a multinomial parameter vectorsuch that

The variational distribution is aswhere is a Gamma distribution,where a is a shape parameter, while b is a scale parameter, is the Gamma function (67). The entropy of (74) is aswhere is the derivative of the log gamma function (digamma function),while is evaluated aswhile and are control parameters for , defined aswhile is defined as

The variational distribution is aswhere and are control parameters for W, defined asand

Given the variational parameters , , and in (78), (79), (81) and (82), the estimates of and are realized by the determination of the Gamma means and [118-121]. It can be verified that the mean in (73), (79) and (80) can be evaluated via (81) and (82) as a mean of a Gamma distributionwhile is aswhere digamma function (76).

The mean in (80) and (82) can be evaluated via (78) and (79), as a mean of a Gamma distributionand is yielded as

As the , and variational distributions are determined via (68), (73) and (80) the evaluation of (59) is straightforward.

Using the defined terms, the term from (57) can be evaluated aswhile the entropy of the variational distribution from (58) can be evaluated as

Thus, from (87) and (88), the lower bound in (57) is as

The next problem is the estimation of the control parameters in (48) assuch that is a basis estimationand is a system estimationsuch that the variational lower bound in (89) is maximized[118-121]. It is achieved for the unitary as follows. The maximization problem can be formalized via the derivative of andwhich is solvable via[118,120]and

After some calculations, and from (90) are asandrespectively.

From (97) and (98), the estimation in (90) is therefore straightforwardly yielded. Therefore, using the parameters and , the optimal variational distributions , and can be substituted to estimate .

Using (97) and (98), the estimation of terms (42), (43) and (55) are yielded asand

The evaluation of (97) and (98) therefore is yielded in an iterative manner through the , , , and , and the K* optimal number of bases, K, is determined with respect to (89) such thatwhere refers to from (89) at a particular base number .

The proof is concluded here. ■

The schematic representation of unitary is depicted in Fig. 3.

Figure 3

Representation of the unitary over a total evolution time , with factored bases and source systems ( in our setting). The factorization is represented by the solid-line arrows. At a given , , the input system of subject of factorization is , . Term is expressed as , where is a unitary, , , which sets a computational basis for , . The basis matrix is with bases, is a Hamiltonian, and , . The factorization decomposes into , and for the total evolution , where , while is as . Terms and are control parameters for and (controlling is depicted by the dashed-line arrows) to evaluate the parameters as and , estimated by and as and .

Quantum constant Q transform

As the basis estimations (99) are determined via (97), the next problem is the partitioning of the bases with respect to , see (8). To achieve the partitioning, first the bases of are transformed by the is the quantum constant transform[128]. The operation is similar to the discrete QFT (quantum Fourier transform) transform[117], and defined in the following manner.

The transform is defined aswhere is a quantum state of the computational basis , and in the current settingandthus is aswhile is selected such thatholds, and is defined via the following relationfrom which is yielded at a given , and , aswhile is a windowing function[129] that localizes the wavefunctions of the quantum register, defined via parameter as

(Footnote: The function in (110) is the so-called Hanning window[129]).

The output states of therefore identify a set of states, asthat formulates an orthonormal basis.

The inverse of will be processed as the partitioning is completed, with the same windowing function, defined as

Applying (103) on the estimated bases yields the transformed bases, aswhere is as,

After the application of (113), the resulting system is therefore aswhere .

Basis partitioning unitary

Theorem 2.

(Partitioning the bases of source systems). The transformed bases can be partitioned to partitions via the partitioning unitary operation.

Proof. As the transforms of the basis estimations (99) are determined via (113), the transformed bases are partitioned to partitions via the unitary operation, as follows.

Let the system state from (115) be denoted byand let be the estimation of [130], defined aswhereis a tensor (multidimensional array)[131,132] with dimension , and sizewhere is the size of the i-th dimension .

Letbe a translation tensor of sizewithasandand letbe a tensor of sizewithasand withasandthusandwhile

The term is evaluated aswhere is the indexing for the elements of the tensor.

Let refer to the j-th column of , and let refer to the j-th lateral slice of . Then, let be a unitary operation that achieves the decomposition of (117) with respect to a given , , aswith a particular cost function of the unitary defined via the quantum relative entropy function, aswhere is the density matrix associated with is as in (116),while is given in (117).

Using (139), the -transformed bases are partitioned into classes, the partition Ω outputted by is evaluated aswhere Q is a size matrix, such that

Since in our setting, the partition (142) can be rewritten aswhere identifies a cluster of Q-transformed bases for m-th system state,ofbases formulated via the base estimations (99) for the m-th system state in (8), such that

Since the partitioning is made over the Q transformed bases, the output of is then transformed by the inverse transformation (112). ■

Inverse quantum constant Q transform

Applying the inverse transformation (112) on the partitions (143) of the Q transformed bases yields the decomposition of the bases of onto classes, asand since where identifies a cluster of bases for m-th system state.

Therefore, the resulting system state is as

The next problem is therefore the evaluation of the estimations of the source systems and , as given in (7) from . Using the system state (150), the system separation is produced by the unitary that realizes the inverse quantum DSTFT (discrete short-time Fourier transform)[129].

Inverse quantum DSTFT and quantum DFT

The result of unitary is evaluated further by the unitary.

Theorem 3.

(Target source system recovery). Source system can be extracted by the and discrete quantum Fourier transform on the output of an HRE quantum memory.

Proof. The inverse quantum DSTFT transformation applied to a state of the computational basisis defined aswhere is selected such thatholds, setformulates an new orthonormal basis, while is a windowing function[129].

Using system state in (150), let be a k-th basis of cluster , and let be defined asand let system identify (33) aswhere is the eigenvector of the Hamiltonian of , is the cardinality of cluster , while .

Since the values are some parameters of , we can redefine (156) aswhereand

In our setting, using as input parameter available from the block, we redefine the formula of (152) via a unitary , aswhere we set to unity,

Thus, applying (160) on (157) yieldswhereand , thus (162) can be rewritten as

As follows, ifthen, the resulting probability iswhile for the remaining j-s, the probabilities are vanished out, thusif

Therefore, applying the discrete quantum Fourier transform on the resulting system state (164), defined in our setting asyields the source system in terms of the K1 bases, asthat identifies the target system from (35).

The proof is concluded here. ■

The state of the quantum register after the operation and after the operation is depicted in Fig. 4.

Figure 4

(a) The state of the quantum register after the operation. The quantum register contains states, , each with probability , with a unit distance between the states (depicted by the red dots). (b) The state of the quantum register after the operation. The quantum register contains quantum states, , , each with probability , with a distance between the states (depicted by the red dots; the vanished-out states of the quantum register are depicted by the black dots).

Retrieval Efficiency

This section evaluates the retrieval efficiency of an HRE quantum memory in terms of the achievable output SNR values.

Theorem 4.

(Retrieval efficiency of an HRE quantum memory). The SNR of the output quantum system of an HRE quantum memory is evolvable from the difference of the wave function energy ratios taken between the input system, the quantum register system, and the output quantum system.

Proof. Let be an arbitrary quantum system fed into the input of an HRE quantum memory unit,and let be the state outputted from the QR quantum register,where U is an unknown transformation.

Let be the output system of as given in (170), that can be rewritten aswhere U is the operator of the integrated unitary operations of the HRE quantum memory, defined as

Then, let be a verification oracle that computes the energy E of a wavefunction [133] aswhere is a Hamiltonian.

Then, let evaluate the corresponding energies of wavefunctions , and via , asand

Then, let Δ be the difference of the ratios of wavefunction energies, defined aswhereand

From the quantities of (176)–(178), let be the SNR of the output system , defined aswherewhile Δ is as given in (179).

Therefore, the SNR of the output system can be evolved from the difference of the ratios of the wavefunction energies as

It also can be verified that Δ from (179) can be rewritten aswhere ΔSNR is an SNR difference, defined as

The high SNR values are reachable at moderate values of wavefunction energy ratio differences (179), therefore a high retrieval efficiency (high SNR values) can be produced by the local unitaries of the memory unit (see also Fig. 5).

Figure 5

Verification of the retrieval efficiency of an HRE quantum memory unit via an verification oracle. In the verification procedure, an unknown quantum system is stored in the quantum register that is evolved by an unknown operation of the quantum register. The output of is an unknown quantum system that is processed further by the integrated unitary operations of the HRE quantum memory. The output system of the HRE quantum memory is (170). The oracle evaluates the SNR of the readout quantum system .

The proof is concluded here. ■

The verification of the retrieval efficiency of the output of an HRE quantum memory unit is depicted in Fig. 5.

The output SNR values in the function of the Δ wave function energy ratio difference are depicted in Fig. 6.

Figure 6

The output SNR values, , of an HRE quantum memory in the function of , where , , , , and .

Conclusions

Quantum memories are a cornerstone of the construction of quantum computers and a high-performance global-scale quantum Internet. Here, we defined the HRE quantum memory for near-term quantum devices. We defined the unitary operations of an HRE quantum memory and proved the learning procedure. We showed that the local unitaries of an HRE quantum memory integrates a group of quantum machine learning operations for the evaluation of the unknown quantum system, and a group of unitaries for the target system recovery. We determined the achievable output SNR values. The HRE quantum memory is a particularly convenient unit for gate-model quantum computers and the quantum Internet.

Ethics statement

This work did not involve any active collection of human data.

Supplemental Information.