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.

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 QR mixed quantum system in QR.
The novel contributions of our manuscript are as follows: 1. We define a novel quantum memory called high-retrieval-efficiency (HRE) quantum memory.
2. 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. 3. 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. 4. We evaluate the retrieval efficiency of an HRE quantum memory and the output SNR. 5. 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 ρ in be an unknown input quantum system formulated by n unknown density matrices, 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 ( = d 2 for a qubit system). For simplicity, we focus on = d 2 dimensional quantum systems throughout the derivations.
The U QR unknown evolution operator of the QR quantum register defines a mixed state σ QR as ∑ σ ρ is an unknown complex quantity, defined as Then, let us rewrite σ QR t ( ) from (3) as QR t in QR t ( ) ( ) where ρ in is as in (1), and ζ QR t ( ) is an unknown residual density matrix at a given t. Therefore, (7) can be expressed as a sum of M source quantum systems, where ρ m is the m-th source quantum system and = … m M 1, , , where in our setting, since where X i m t ( , ) is a complex quantity associated with an m-th source system, ) , and The aim is to find the V QG inverse matrix of the unknown evolution matrix U QR in (2), as QG QG 1 that yields the separated readout quantum system of the HRE quantum memory unit for = … t T 1, , , such that for a given t, For a total evolution time T, the target σ out density matrix is yielded at the output of the HRE quantum memory unit, as out 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 QR 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, = … m M 1, , , such that = m 1 identifies the unknown input quantum system stored in the quantum register (target source system), while = … m M 2, , 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. 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

Integrated Local Unitaries
This section defines the local unitary operations integrated on an HRE quantum memory unit.
Quantum machine learning unitary. The U ML quantum machine learning unitary implements an unsupervised learning for a blind separation of the unlabeled quantum register. The U ML unitary is defined as Proof. The aim of the U F factorization unitary is to factorize the mixed quantum register (2) into a basis matrix U B and a quantum system ρ → W , as where U B is a complex basis matrix, defined as Figure 2. Detailed procedures of an HRE quantum memory. The unknown input quantum system is stored in the QR quantum register that realizes an unknown transformation. The density matrix of the quantum register is the sum of = M 2 source systems, where source system = m 1 identifies the valuable unknown input quantum system stored in the quantum register, while = m 2 identifies an unknown undesired residual quantum system. The U F unitary evaluates K bases for the source system and defines a W auxiliary quantum system. The U CQT unitary is a preliminary operation for the partitioning of the K bases onto M clusters via unitary U P . The U P unitary regroups the bases with respect to the = M 2 source systems. The results are then processed by the ∼ † U DSTFT and U DFT unitaries to extract the source system = m 1 on the output of the memory unit.
www.nature.com/scientificreports www.nature.com/scientificreports/ The first part of the problem is therefore to find (22), where u mk is a unitary that sets a computational basis for where H mk is a Hamiltonian, as where X m t ( , ) is defined in (14).
As follows, for the total evolution time T,  → ∈ × X M T can be defined as Thus, by applying of the u mk unitaries for the total evolution time T, where K m is the number of bases associated with the m-th source system, . In our setting = M 2, and our aim is to get the system state = m 1 on the output of the HRE quantum memory, thus a Φ ⁎ target output system state is defined as 1 and let Then, let ρ→ X be a density matrix associated with → X , defined as and let be the density matrix associated with (36). The aim of the estimation is to minimize the ⋅ ⋅ D( ) quantum relative entropy function taken between ρ→ X and ρ ∼ X , thus an f U ( ) F objective function for U F is defined via (37) and (38) as (41), a factorization method is defined for U F that is based on the fundamentals of Bayesian nonnegative matrix factorization [118][119][120][121][122][123][124][125][126][127] (Footnote: The U F 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 ⋅ L( ) likelihood function and the exponential distribution for the control parameters 118-121 α mk and β kt defined for the controlling of u mk and w kt .
Let u mk and w kt from (29) be defined via the control parameters α mk and β kt as exponential distributions can be defined as The problem is therefore can be reduced to determine the model parameters B that are treated as latent variables for the estimation of the control parameters 118-121,125-127 where ⋅ D( ) 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 The quantity in (54) can be estimated via (42) and (43) as ) in (29) can be rewritten as However, since the exact solution does not exists [118][119][120][121] , since it would require the factorization of are unknown. This problem can be solved by a variational Bayesian inference procedure [118][119][120][121][125][126][127] , via the maximization of the lower bound of a likelihood function is a joint variational distribution, as can be approximated as [118][119][120][121] (57) is related to (50) as The result in (59) therefore also determines the number K of bases selected for the factorization unitary U F .
is obtainable, the variational distributions have to be evaluated as a a a for some functions f a ( ) and g a ( ), and a a for some constant b, (note: for simplicity, we use  ⋅ ( ) for the expectation function), while where x t 0 1 By utilizing a variational Poisson-Exponential Bayesian learning [118][119][120][121] , these variational distributions can be evaluated as follows. The where M is a multinomial distribution, while η mkt is a multinomial parameter   such that where a is a shape parameter, while b is a scale parameter, ⋅ Γ f ( ) is the Gamma function (67). The entropy of (74) is as is the derivative of the log gamma function (digamma function),    (78), (79), (81) and (82), the estimates of U B and W are realized by the determination of the Gamma means  u ( ) mk and  w ( ) kt [118][119][120][121] . It can be verified that the mean  w ( ) kt in (73), (79) and (80) can be evaluated via (81) and (82) as a mean of a Gamma distribution (80) and (82) can be evaluated via (78) and (79), as a mean of a Gamma distribution (68), (73) and (80) the evaluation of (59) is straightforward.
Using the defined terms, the term  (57) can be evaluated as     (48) and F kt is a system estimation kt kt such that the variational lower bound L D v in (89) is maximized [118][119][120][121] . It is achieved for the unitary U F as follows.
The maximization problem can be formalized via the ∂ L D mk mk mk mk  After some calculations, E mk and F kt from (90) are as   (97) and (98), the estimation of terms u k (42), w kt (43) and κ kt (55) are yielded as The evaluation of (97) and (98) therefore is yielded in an iterative manner through the α and η mkt , and the K* optimal number of bases, K, is determined with respect to (89) such that (89) at a particular base number K. The proof is concluded here. ■ The schematic representation of unitary U F is depicted in Fig. 3. (97), the next problem is the partitioning of the K bases with respect to M, see (8). To achieve the partitioning, first the bases of U B are transformed by the U CQT is the quantum constant Q transform 128 . The U CQT operation is similar to the discrete QFT (quantum Fourier transform) transform 117 , and defined in the following manner.
The U CQT transform is defined as Figure 3. Representation of the U F unitary over a total evolution time t, with K factored bases and M source systems ( = M 2 in our setting). The factorization is represented by the solid-line arrows. At a given t, , and for the total evolution 1 , while κ kt is as κ = u w mkt m k kt . Terms α mk and β kt are control parameters for u mk and w kt (controlling is depicted by the dashed-line arrows) to evaluate the parameters as α where k is a quantum state of the computational basis B, and in the current setting mk thus B is as while h is selected such that holds, and Q is defined via the following relation from which Q is yielded at a given h, k and K, as is a windowing function 129 that localizes the wavefunctions of the quantum register, defined via parameter h as The function in (110) is the so-called Hanning window 129 ). The φ k output states of U CQT therefore identify a set φ S of states, as k that formulates an orthonormal basis. The † U CQT inverse of U CQT will be processed as the U P partitioning is completed, with the same ⋅ f ( ) W windowing function, defined as where C mk is as,

mk CQT mk
After the application of (113), the resulting system is therefore as

Basis partitioning unitary. Theorem 2. (Partitioning the bases of source systems). The Q transformed bases can be partitioned to M partitions via the U P partitioning unitary operation.
Proof. As the U CQT transforms of the E { } mk basis estimations (99) are determined via C B (113), the Q transformed bases are partitioned to M partitions via the U P unitary operation, as follows.
Let the system state from (115) be denoted by www.nature.com/scientificreports www.nature.com/scientificreports/ where is a tensor (multidimensional array) 131,132 with dimension T dim( ), and size be a translation tensor of size 3 and let be a tensor of size The term RE is evaluated as ) is the indexing for the elements of the tensor. Let ∀ E m k ( , ) refer to the j-th column of E, and let ∀ H k t (1, , ) refer to the j-th lateral slice of H. Then, let be a U P unitary operation that achieves the decomposition of (117) with respect to a given k, = … k K 1, , , as k with a particular cost function f U ( ) P of the U P unitary defined via the quantum relative entropy function, as where ρ S is the density matrix associated with S is as in (116), while  S is given in (117). Using (139), the Q-transformed bases are partitioned into M classes, the partition Ω outputted by U P is evaluated as k m M 1 Since = M 2 in our setting, the partition (142) can be rewritten as where Ω Q m ( ) identifies a cluster of K m Q-transformed bases for m-th system state, Since the partitioning is made over the Q transformed bases, the output of U P is then transformed by the † U CQT inverse transformation (112).
(1) ( 2) where γ m ( ) identifies a cluster of K m bases for m-th system state. Therefore, the resulting system state is as CQT P B

CQT P CQT B
The next problem is therefore the evaluation of the estimations of the = M 2 source systems ρ in and ζ QR t ( ) , as given in (7) from χW. Using the system state (150), the system separation is produced by the † U DSTFT unitary that realizes the inverse quantum DSTFT (discrete short-time Fourier transform) 129 . Proof. The † U DSTFT inverse quantum DSTFT transformation applied to a state k of the computational basis

Inverse quantum DSTFT and quantum DFT. The result of unitary
is defined as  Since the k 1 values are some parameters of U ML , we can redefine (156) as In our setting, using = k m 1 as input parameter available from the U ML block, we redefine the formula of (152) via a unitary ∼ † U DSTFT , as  As follows, if    The proof is concluded here. ■ The state of the QR quantum register after the ∼ † U CQT operation and after the ∼ † U DSTFT operation is depicted in Fig. 4.

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 ψ in be an arbitrary quantum system fed into the input of an HRE quantum memory unit, in i i and let φ be the state outputted from the QR quantum register,

QR in
where U QG is an unknown transformation.
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 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).
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.

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.

Data availability
This work does not have any experimental data. In the verification procedure, an unknown quantum system ψ is stored in the QR quantum register that is evolved by an unknown operation U QR of the QR quantum register. The output of QR is an unknown quantum system φ that is processed further by the U integrated unitary operations of the HRE quantum memory. The output system of the HRE quantum memory is Φ ⁎ (170). The O O V oracle evaluates the SNR of the readout quantum system Φ ⁎ .