Abstract
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.
Similar content being viewed by others
Introduction
Quantum memories are a fundamental of any global-scale quantum Internet1,2,3,4,5,6. However, while quantum repeaters can be realized without the necessity of quantum memories1,3, these units, in fact, are required for guaranteeing an optimal performance in any high-performance quantum networking scenario3,4,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32. Therefore, the utilization of quantum memories still represents a fundamental problem in the quantum Internet33,34,35,36,37,38,39,40,41,42, since the near-term quantum devices (such as quantum repeaters5,6,8,32,43,44,45,46,47) and gate-model quantum computers48,49,50,51,52,53,54,55,56,57,58,59 have to store the quantum states in their local quantum memories43,44,45,46,47,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84. 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 scenario4,5,7,8,9,11,12. The optimization of the readout procedure is therefore a hard and complex problem. Several physical implementations have been developed in the last few years85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105. 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 communication106,107,108,109 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 efficiency110,111,112. 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 σQR 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 \({\rho }_{in}\) be an unknown input quantum system formulated by n unknown density matrices,
where \({\lambda }_{i}^{(in)}\ge 0\), and \({\sum }_{i=1}^{n}\,{\lambda }_{i}^{(in)}=1\).
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 UQR unknown evolution operator of the QR quantum register defines a mixed state σQR as
where \({\lambda }_{i}\ge 0\), \({\sum }_{i=1}^{n}\,{\lambda }_{i}=1\).
Let us allow to rewrite (2) for a particular time t, \(t=1,\ldots ,T\), where T is a total evolution time, via a mixed system \({\sigma }_{QR}^{(t)}\), as
where \({U}_{QR}^{(t)}\) is an unknown evolution matrix of the QR quantum register at a given t, with a dimension
with \(0\le {\lambda }_{i}^{(t)}\le 1\), \({\sum }_{i}\,{\lambda }_{i}^{(t)}=1\), while \({X}_{i}^{(t)}\in {\mathbb{C}}\) is an unknown complex quantity, defined as
and
Then, let us rewrite \({\sigma }_{QR}^{(t)}\) from (3) as
where \({\rho }_{in}\) is as in (1), and \({\zeta }_{QR}^{(t)}\) is an unknown residual density matrix at a given t.
Therefore, (7) can be expressed as a sum of M source quantum systems,
where \({\rho }_{m}\) is the m-th source quantum system and \(m=1,\ldots ,M\), where
in our setting, since
and
In terms of the M subsystems, (3) can be rewritten as
where \({X}_{i}^{(m,t)}\) is a complex quantity associated with an m-th source system,
with \(0\le {\lambda }_{i}^{(m,t)}\le 1\), \({\sum }_{m}\,{\sum }_{i}\,{\lambda }_{i}^{(m,t)}=1\), and
The aim is to find the VQG inverse matrix of the unknown evolution matrix UQR in (2), as
that yields the separated readout quantum system of the HRE quantum memory unit for \(t=1,\ldots ,T\), such that for a given t,
where
For a total evolution time T, the target σout density matrix is yielded at the output of the HRE quantum memory unit, as
with 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 UQR 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, \(m=1,\ldots ,M\), such that \(m=1\) identifies the unknown input quantum system stored in the quantum register (target source system), while \(m=2,\ldots ,M\) 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, UML, 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.
The procedures realized by the integrated unitary operations of the HRE quantum memory are depicted in Fig. 2.
Experimental implementation
An experimental implementation of an HRE quantum memory in a near-term quantum device52 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 devices52,53,54,55,56, such as superconducting units53. The application of a HRE quantum memory in a quantum Internet setting1,2,4,5,6 can be implemented via noisy quantum links between the quantum repeaters8,32,43,44,45,46,47 (e.g., optical fibers7,62,113, wireless quantum channels27,28, free-space optical channels114) and fundamental quantum transmission protocols24,115,116,117.
Integrated Local Unitaries
This section defines the local unitary operations integrated on an HRE quantum memory unit.
Quantum machine learning unitary
The UML quantum machine learning unitary implements an unsupervised learning for a blind separation of the unlabeled quantum register. The UML unitary is defined as
where UF is a factorization unitary, UCQT is the quantum constant Q transform, UP is a partitioning unitary, while \({U}_{CQT}^{\dagger }\) is the inverse of \({U}_{CQT}\).
Factorization unitary
Theorem 1.
(Factorization of the unknown mixed quantum system of the quantum register). The UF unitary factorizes the unknown \({\sigma }_{QR}\) mixed quantum system of the QR quantum register into a unitary \({u}_{mk}={e}^{-i{H}_{mk}\tau /\hslash }\), with a Hamiltonian \({H}_{mk}\) and application time \(\tau \), and into a system \({w}_{kt}\), where \(t=1,\ldots ,T\), \(m=1,\ldots ,M\), and \(k=1,\ldots ,K\), and where T is the evolution time, M is the number of source systems of \({\sigma }_{QR}\), and K is the number of bases.
Proof. The aim of the UF factorization unitary is to factorize the mixed quantum register (2) into a basis matrix UB and a quantum system \({\overrightarrow{\rho }}_{W}\), as
where UB is a complex basis matrix, defined as
and \({\overrightarrow{\rho }}_{W}\in {{\mathbb{C}}}^{K\times T}\) is a complex matrix, defined as
where
where \(0\le {v}_{k}^{(t)}\le 1\), and \({\sum }_{k=1}^{K}\,{v}_{k}^{(t)}=1\), while \(K\) is the total number of bases of \({U}_{B}\), while \({W}_{k}^{(t)}\in {\mathbb{C}}\) is a complex quantity, as
The first part of the problem is therefore to find (22), where \({u}_{mk}\) is a unitary that sets a computational basis for \({W}_{k}^{(t)}\) in (25), defined as
where Hmk is a Hamiltonian, as
where Gmk is the eigenvalue of basis \(|{k}_{m}\rangle \), \({H}_{mk}|{k}_{m}\rangle ={G}_{mk}|{k}_{m}\rangle \), while \(\tau \) is the application time of umk.
The second part of the problem is to determine W, as
where \({W}_{k}^{(t)}={w}_{kt}\) is a system state, that formulates \({\tilde{X}}^{(m,t)}\) as
where \({\tilde{X}}^{(m,t)}\) is an approximation of \({X}^{(m,t)}\),
where \({X}^{(m,t)}\) is defined in (14).
As follows, for the total evolution time T, \(\overrightarrow{X}\in {{\mathbb{C}}}^{M\times T}\) can be defined as
and the challenge is to evaluate (31) as a decomposition
Thus, by applying of the umk unitaries for the total evolution time T, \(\tilde{X}\in {{\mathbb{C}}}^{M\times T}\) is as
where Km is the number of bases associated with the m-th source system,
and \(0\le {|{\ell }_{m,k}(\tau )|}^{2}\le 1\), \({\sum }_{m=1}^{M}\,{\sum }_{k=1}^{K}\,{|{\ell }_{m,k}(\tau )|}^{2}=1\).
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 \(|{\Phi }^{\ast }\rangle \) target output system state is defined as
where K1 is the number of bases for source system \(m=1\), \({k}_{1}=1,\ldots ,{K}_{1}\).
Let rewrite the system state \(\tilde{X}\) (32) as
and let
and
Then, let \({\rho }_{\overrightarrow{X}}\) be a density matrix associated with \(\overrightarrow{X}\), defined as
and let
be the density matrix associated with (36).
The aim of the estimation is to minimize the \(D(\cdot \parallel \cdot )\) quantum relative entropy function taken between \({\rho }_{\overrightarrow{X}}\) and \({\rho }_{\tilde{X}}\), thus an \(f({U}_{F})\) objective function for \({U}_{F}\) is defined via (37) and (38) as
To achieve the objective function \(f({U}_{F})\) in (41), a factorization method is defined for \({U}_{F}\) that is based on the fundamentals of Bayesian nonnegative matrix factorization118,119,120,121,122,123,124,125,126,127 (Footnote: The \({U}_{F}\) factorization unitary applied on the mixed state of the quantum register is analogous to a Poisson-Exponential Bayesian nonnegative matrix factorization118,119,120,121 process). The method adopts the Poisson distribution as \( {\mathcal L} (\,\cdot \,)\) likelihood function and the exponential distribution for the control parameters118,119,120,121 \({\alpha }_{mk}\) and \({\beta }_{kt}\) defined for the controlling of \({u}_{mk}\) and \({w}_{kt}\).
Let \({u}_{mk}\) and \({w}_{kt}\) from (29) be defined via the control parameters \({\alpha }_{mk}\) and \({\beta }_{kt}\) as exponential distributions
with mean \({\alpha }_{mk}^{-1}\), and
with mean \({\beta }_{kt}^{-1}\).
Using (41), (42) and (43), a \( {\mathcal L} (\,\cdot \,)\) log likelihood function
can be defined as
thus the objective function \(f({U}_{F})\) can be rewritten via as (45)
The problem is therefore can be reduced to determine the model parameters
that are treated as latent variables for the estimation of the control parameters118,119,120,121,125,126,127
A maximum likelihood estimation \(\tilde{\zeta }\) of (47) is as
where \( {\cal{D}} (\,\cdot \,)\) 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 \( {\mathcal L} (\overrightarrow{X}|\zeta )\) as
where \(\overrightarrow{\kappa }\) is a complex matrix, \(\overrightarrow{\kappa }\in {{\mathbb{C}}}^{M\times T}\),
where
with
where
The quantity in (54) can be estimated via (42) and (43) as
Using (54), \({\tilde{X}}^{(m,t)}\) in (29) can be rewritten as
However, since the exact solution does not exists118,119,120,121, since it would require the factorization of \( {\cal{D}} (\overrightarrow{\kappa },{U}_{B},W|\overrightarrow{X},\zeta )\), such that \(\zeta ,{U}_{B},W\) are unknown.
This problem can be solved by a variational Bayesian inference procedure118,119,120,121,125,126,127, via the maximization of the lower bound of a likelihood function \({ {\mathcal L} }_{{ {\cal{D}} }_{v}}\)
where \({ {\cal{D}} }_{v}\) is a variational distribution, while \(H({ {\cal{D}} }_{v}(\overrightarrow{\kappa },{U}_{B},W))\) is the entropy of variational distribution \({ {\cal{D}} }_{v}(\overrightarrow{\kappa },{U}_{B},W)\),
and where \({ {\cal{D}} }_{v}(\overrightarrow{\kappa },{U}_{B},W)\) is a joint variational distribution, as
from which distribution \( {\cal{D}} (\overrightarrow{\kappa },{U}_{B},W|\overrightarrow{X},\zeta )\) can be approximated as118,119,120,121
The function \({ {\mathcal L} }_{{ {\cal{D}} }_{v}}\) in (57) is related to (50) as
The result in (59) therefore also determines the number \(K\) of bases selected for the factorization unitary \({U}_{F}\). The \({ {\cal{D}} }_{v}\) variational distributions \({ {\cal{D}} }_{v}({\kappa }_{mkt})\), \({ {\cal{D}} }_{v}({u}_{k})\) and \({ {\cal{D}} }_{v}({w}_{kt})\) are determined for the unitary UF as follows.
Let \({ {\cal{D}} }_{v}(\Phi )\) refer to the variational distribution of a given \(\Phi \),
Since only the joint (posterior) distribution \( {\cal{D}} (\overrightarrow{X},\overrightarrow{\kappa },{U}_{B},W|\zeta )\) is obtainable, the variational distributions have to be evaluated as
where \({{\mathbb{E}}}_{{ {\cal{D}} }_{v}(i\ne \Phi )}(\,\cdot \,)\) is the expectation function of the \({ {\cal{D}} }_{v}(i)\) variational distribution of i, such that \(i\ne \Phi \), where \(\Phi \) is as in (62), with
for some functions \(f(a)\) and \(g(a)\), and
for some constant b, (note: for simplicity, we use \({\mathbb{E}}(\,\cdot \,)\) for the expectation function), while
where \({f}_{\delta }(\,\cdot \,)\) is the Dirac delta function, while \({f}_{\Gamma }(\,\cdot \,)\) is the Gamma function,
By utilizing a variational Poisson–Exponential Bayesian learning118,119,120,121, these variational distributions can be evaluated as follows.
The \({ {\cal{D}} }_{v}({\kappa }_{mkt})\) variational distribution is as
where \( {\mathcal M} \) is a multinomial distribution, while \({\eta }_{mkt}\) is a multinomial parameter
while the \({ {\cal{D}} }_{v}({\kappa }^{(m,t)})\) variational distribution is as
where \({\eta }_{k}^{(m,t)}\) is a multinomial parameter vector
such that
The \({ {\cal{D}} }_{v}({u}_{mk})\) variational distribution is as
where \({\mathscr G}(\,\cdot \,)\) is a Gamma distribution,
where a is a shape parameter, while b is a scale parameter, \({f}_{\Gamma }(\,\cdot \,)\) is the Gamma function (67). The entropy of (74) is as
where \({\partial }_{{{\mathscr G}}_{\log }}(\,\cdot \,)\) is the derivative of the log gamma function (digamma function),
while \({\mathbb{E}}({\kappa }_{mkt})\) is evaluated as
while \({\tilde{\alpha }}_{mk}(A)\) and \({\tilde{\alpha }}_{mk}(B)\) are control parameters for \({U}_{B}\), defined as
while \({\tilde{\alpha }}_{mk}(B)\) is defined as
The \({ {\cal{D}} }_{v}({w}_{kt})\) variational distribution is as
where \({\tilde{\beta }}_{kt}(A)\) and \({\tilde{\beta }}_{kt}(B)\) are control parameters for W, defined as
and
Given the variational parameters \({\tilde{\alpha }}_{mk}(A)\), \({\tilde{\alpha }}_{mk}(B)\), \({\tilde{\beta }}_{kt}(A)\) and \({\tilde{\beta }}_{kt}(B)\) in (78), (79), (81) and (82), the estimates of \({U}_{B}\) and \(W\) are realized by the determination of the Gamma means \({\mathbb{E}}({u}_{mk})\) and \({\mathbb{E}}({w}_{kt})\)118,119,120,121. It can be verified that the mean \({\mathbb{E}}({w}_{kt})\) in (73), (79) and (80) can be evaluated via (81) and (82) as a mean of a Gamma distribution
while \({\mathbb{E}}(\log \,{w}_{kt})\) is as
where \({\partial }_{{{\mathscr G}}_{\log }}(\,\cdot \,)\) digamma function (76).
The mean \({\mathbb{E}}({u}_{mk})\) in (80) and (82) can be evaluated via (78) and (79), as a mean of a Gamma distribution
and \({\mathbb{E}}(\log \,{u}_{mk})\) is yielded as
As the \({ {\cal{D}} }_{v}({\kappa }_{mkt})\), \({ {\cal{D}} }_{v}({u}_{mk})\) and \({ {\cal{D}} }_{v}({w}_{kt})\) variational distributions are determined via (68), (73) and (80) the evaluation of (59) is straightforward.
Using the defined terms, the term \({\mathbb{E}}(\log \, {\cal{D}} (\overrightarrow{X},\overrightarrow{\kappa },{U}_{B},W|\zeta ))\) from (57) can be evaluated as
while the \(H({ {\cal{D}} }_{v}(\overrightarrow{\kappa },{U}_{B},W))\) entropy of the variational distribution from (58) can be evaluated as
Thus, from (87) and (88), the lower bound \({ {\mathcal L} }_{{ {\cal{D}} }_{v}}\)in (57) is as
The next problem is the \({\tilde{\tau }}_{k}^{(t)}\) estimation of the control parameters \({\alpha }_{mk},{\beta }_{kt}\) in (48) as
such that \({E}_{mk}\) is a basis estimation
and \({F}_{kt}\) is a system estimation
such that the variational lower bound \({ {\mathcal L} }_{{ {\cal{D}} }_{v}}\) in (89) is maximized118,119,120,121. It is achieved for the unitary \({U}_{F}\) as follows. The maximization problem can be formalized via the \(\partial ({ {\mathcal L} }_{{ {\cal{D}} }_{v}})\) derivative of \({ {\mathcal L} }_{{ {\cal{D}} }_{v}}\)
and
and
After some calculations, \({E}_{mk}\) and \({F}_{kt}\) from (90) are as
and
respectively.
From (97) and (98), the \({\tilde{\tau }}_{mk}^{(t)}\) estimation in (90) is therefore straightforwardly yielded. Therefore, using the parameters \({\tilde{\alpha }}_{mk}(B),{\tilde{\alpha }}_{mk}(A),{\tilde{\beta }}_{kt}(A),{\tilde{\beta }}_{kt}(B)\) and \({\eta }_{mkt}\), the optimal variational distributions \({ {\cal{D}} }_{v}({\kappa }_{mkt})\), \({ {\cal{D}} }_{v}({u}_{mk})\) and \({ {\cal{D}} }_{v}({w}_{kt})\) can be substituted to estimate \({\tilde{\tau }}_{mk}^{(t)}\).
Using (97) and (98), the estimation of terms \({u}_{k}\) (42), \({w}_{kt}\) (43) and \({\kappa }_{kt}\) (55) are yielded as
and
The evaluation of (97) and (98) therefore is yielded in an iterative manner through the \({\tilde{\alpha }}_{mk}(B)\), \({\tilde{\alpha }}_{mk}(A)\), \({\tilde{\beta }}_{kt}(A)\), \({\tilde{\beta }}_{kt}(B)\) and \({\eta }_{mkt}\), and the K* optimal number of bases, K, is determined with respect to (89) such that
where \({ {\mathcal L} }_{{ {\cal{D}} }_{v}}(K)\) refers to \({ {\mathcal L} }_{{ {\cal{D}} }_{v}}\) from (89) at a particular base number \(K\).
The proof is concluded here. ■
The schematic representation of unitary \({U}_{F}\) is depicted in Fig. 3.
Quantum constant Q transform
As the \(\{{\tilde{u}}_{mk}\}\) basis estimations (99) are determined via \(\{{E}_{mk}\}\) (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 \(Q\) transform128. The \({U}_{CQT}\) operation is similar to the discrete QFT (quantum Fourier transform) transform117, and defined in the following manner.
The \({U}_{CQT}\) transform is defined as
where \(|k\rangle \) is a quantum state of the computational basis \(B\), and in the current setting
and
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
while \({f}_{W}(\,\cdot \,)\) is a windowing function129 that localizes the wavefunctions of the quantum register, defined via parameter \(h\) as
(Footnote: The function in (110) is the so-called Hanning window129).
The \(|{\varphi }_{k}\rangle \) output states of \({U}_{CQT}\) therefore identify a set \({\mathscr S}_{\varphi }\) of states, as
that formulates an orthonormal basis.
The \({U}_{CQT}^{\dagger }\) inverse of \({U}_{CQT}\) will be processed as the \({U}_{P}\) partitioning is completed, with the same \({f}_{W}(\,\cdot \,)\) windowing function, defined as
Applying (103) on the \(K\) estimated bases \(\{{E}_{mk}\}\) yields the \({C}_{B}\) transformed bases, as
where \({C}_{mk}\) is as,
After the application of (113), the resulting system is therefore as
where \({C}_{B}W\in {{\mathbb{C}}}^{M\times T}\).
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
and let \(\tilde{S}\) be the estimation of \(S\)130, defined as
where
is a tensor (multidimensional array)131,132 with dimension \({\rm{\dim }}({\mathscr T})\), and size
where \(|{d}_{i}({\mathscr T})|\) is the size of the i-th dimension \({d}_{i}({\mathscr T})\).
Let
be a translation tensor of size
with
as
and
and let
be a tensor of size
with
as
and with
as
and
thus
and
while
The term \(\langle {\mathcal R} {\mathcal E} \rangle \) is evaluated as
where \( {\mathcal R} (i,j)\) is the indexing for the elements of the tensor.
Let \( {\mathcal E} (\forall m,k)\) refer to the j-th column of \( {\mathcal E} \), and let \( {\mathcal H} (1,k,\forall t)\) refer to the j-th lateral slice of \( {\mathcal H} \). Then, let be a \({U}_{P}\) unitary operation that achieves the decomposition of (117) with respect to a given \(k\), \(k=1,\ldots ,K\), as
with a particular cost function \(f({U}_{P})\) of the \({U}_{P}\) unitary defined via the quantum relative entropy function, as
where \({\rho }_{S}\) is the density matrix associated with \(S\) is as in (116),
while \(\tilde{S}\) 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
where Q is a \(1\times K\) size matrix, such that
Since \(M=2\) in our setting, the partition (142) can be rewritten as
where \({\Omega }_{Q}^{(m)}\) identifies a cluster of \({K}_{m}\) Q-transformed bases for m-th system state,
of
bases 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 \({U}_{P}\) is then transformed by the \({U}_{CQT}^{\dagger }\) inverse transformation (112). ■
Inverse quantum constant Q transform
Applying the \({U}_{CQT}^{\dagger }\) inverse transformation (112) on the partitions (143) of the Q transformed bases yields the decomposition of the bases of \({U}_{B}\) onto \(M\) classes, as
and since \(M=2\)
where \({\gamma }^{(m)}\) identifies a cluster of \({K}_{m}\) 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 \(M=2\) source systems \({\rho }_{in}\) and \({\zeta }_{QR}^{(t)}\), as given in (7) from \(\chi W\). Using the system state (150), the system separation is produced by the \({U}_{{\rm{DSTFT}}}^{\dagger }\) unitary that realizes the inverse quantum DSTFT (discrete short-time Fourier transform)129.
Inverse quantum DSTFT and quantum DFT
The result of unitary \({U}_{ML}\) is evaluated further by the \({U}_{{\rm{DSTFT}}}^{\dagger }\) unitary.
Theorem 3.
(Target source system recovery). Source system \(m=1\) can be extracted by the \({U}_{{\rm{DSTFT}}}^{\dagger }\) and \({U}_{DFT}\) discrete quantum Fourier transform on the output of an HRE quantum memory.
Proof. The \({U}_{{\rm{DSTFT}}}^{\dagger }\) inverse quantum DSTFT transformation applied to a state \(|k\rangle \) of the computational basis
is defined as
where \(h\) is selected such that
holds, set
formulates an new orthonormal basis, while \({f}_{W}(\,\cdot \,)\) is a windowing function129.
Using system state \(\chi W\) in (150), let \({\gamma }^{(m,k)}\) be a k-th basis of cluster \({\gamma }^{(m)}\), and let \({(\chi W)}^{(m,t)}\) be defined as
and let system \(|\chi W\rangle \) identify (33) as
where \(|{k}_{m}\rangle \) is the eigenvector of the Hamiltonian of \({\gamma }^{(m,{k}_{m})}\), \({K}_{m}\) is the cardinality of cluster \({\gamma }^{(m)}\), while \({\sum }_{m\mathrm{=1}}^{M}{\sum }_{{k}_{m}\mathrm{=1}}^{{K}_{m}}\alpha \mathrm{=1}\).
Since the \(|{k}_{1}\rangle \) values are some parameters of \({U}_{ML}\), we can redefine (156) as
where
and
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 \({\tilde{U}}_{{\rm{DSTFT}}}^{\dagger }\), as
where we set \({f}_{W}(j-h)\) to unity,
Thus, applying (160) on (157) yields
where
and \({\sum }_{j=0}^{K-1}\,{e}^{-2\pi ij{x}_{m,{k}_{m}}/K}=1\), thus (162) can be rewritten as
As follows, if
then, the resulting \({\rm{\Pr }}(j)\) probability is
while for the remaining j-s, the probabilities are vanished out, thus
if
Therefore, applying the \({U}_{DFT}\) discrete quantum Fourier transform on the resulting system state (164), defined in our setting as
yields the source system \(m=1\) in terms of the K1 bases, as
that identifies the target system from (35).
The proof is concluded here. ■
The state of the \(QR\) quantum register after the \({\tilde{U}}_{{\rm{CQT}}}^{\dagger }\) operation and after the \({\tilde{U}}_{{\rm{DSTFT}}}^{\dagger }\) 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 \(|{\psi }_{in}\rangle \) be an arbitrary quantum system fed into the input of an HRE quantum memory unit,
and let \(|\varphi \rangle \) be the state outputted from the QR quantum register,
where UQG is an unknown transformation.
Let \(|{\Phi }^{\ast }\rangle \) be the output system of as given in (170), that can be rewritten as
where U is the operator of the integrated unitary operations of the HRE quantum memory, defined as
Then, let \({{\mathcal O}}_{V}\) be a verification oracle that computes the energy E of a wavefunction \(|\psi \rangle ={\sum }_{i}\,{c}_{i}|{\phi }_{i}\rangle \)133 as
where \(\hat{H}\) is a Hamiltonian.
Then, let evaluate the corresponding energies of wavefunctions \(|{\psi }_{in}\rangle \), \(|\varphi \rangle \) and \(|{\Phi }^{\ast }\rangle \) via \({{\mathcal O}}_{V}\), as
and
Then, let Δ be the difference of the ratios of wavefunction energies, defined as
where
and
From the quantities of (176)–(178), let \({\rm{SNR}}(|{\Phi }^{\ast }\rangle )\) be the SNR of the output system \(|{\Phi }^{\ast }\rangle \), defined as
where
while 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 as
where 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).
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.
References
Pirandola, S. & Braunstein, S. L. Unite to build a quantum internet. Nature 532, 169–171 (2016).
Lloyd, S. et al. Infrastructure for the quantum Internet. ACM SIGCOMM Computer Communication Review 34, 9–20 (2004).
Pirandola, S. End-to-end capacities of a quantum communication network. Commun. Phys. 2, 51 (2019).
Wehner, S., Elkouss, D. & Hanson, R. Quantum internet: A vision for the road ahead. Science 362, 6412 (2018).
Van Meter, R. Quantum Networking. ISBN 1118648927, 9781118648926, John Wiley and Sons Ltd (2014).
Kimble, H. J. The quantum Internet. Nature 453, 1023–1030 (2008).
Van Meter, R., Ladd, T. D., Munro, W. J. & Nemoto, K. System Design for a Long-Line Quantum Repeater. IEEE/ACM Transactions on Networking 17(3), 1002–1013 (2009).
Van Meter, R., Satoh, T., Ladd, T. D., Munro, W. J. & Nemoto, K. Path selection for quantum repeater networks. Networking Science 3(1–4), 82–95 (2013).
Van Meter, R. & Devitt, S. J. Local and Distributed Quantum Computation. IEEE Computer 49(9), 31–42 (2016).
Pirandola, S., Laurenza, R., Ottaviani, C. & Banchi, L. Fundamental limits of repeaterless quantum communications. Nature Communications, 15043, https://doi.org/10.1038/ncomms15043 (2017).
Pirandola, S. et al. Theory of channel simulation and bounds for private communication. Quantum Sci. Technol. 3, 035009 (2018).
Pirandola, S. Capacities of repeater-assisted quantum communications. Quantum Sci. Technol, 4, 045006 (2019).
Gyongyosi, L. & Imre, S. Decentralized Base-Graph Routing for the Quantum Internet. Physical Review A, American Physical Society, https://doi.org/10.1103/PhysRevA.98.022310 (2018).
Gyongyosi, L. & Imre, S. Dynamic topology resilience for quantum networks. Proc. SPIE 10547, Advances in Photonics of Quantum Computing, Memory, and Communication XI, 105470Z, https://doi.org/10.1117/12.2288707 (2018).
Gyongyosi, L. & Imre Topology Adaption for the Quantum Internet. Quantum Information Processing. https://doi.org/10.1007/s11128-018-2064-x (2018). Springer Nature.
Gyongyosi, L. & Imre, S. Entanglement Access Control for the Quantum Internet. Quantum Information Processing, Springer Nature, https://doi.org/10.1007/s11128-019-2226-5 (2019).
Gyongyosi, L. & Imre, S. Opportunistic Entanglement Distribution for the Quantum Internet, Scientific Reports, Nature, https://doi.org/10.1038/s41598-019-38495-w (2019).
Gyongyosi, L. & Imre, S. Adaptive Routing for Quantum Memory Failures in the Quantum Internet, Quantum Information Processing, Springer Nature, https://doi.org/10.1007/s11128-018-2153-x (2018).
Quantum Internet Research Group (QIRG), web: https://datatracker.ietf.org/rg/qirg/about/ (2018).
Laurenza, R. & Pirandola, S. General bounds for sender-receiver capacities in multipoint quantum communications. Phys. Rev. A 96, 032318 (2017).
Gyongyosi, L. & Imre, S. Multilayer Optimization for the Quantum Internet, Scientific Reports, Nature, https://doi.org/10.1038/s41598-018-30957-x (2018).
Gyongyosi, L. & Imre, S. Entanglement Availability Differentiation Service for the Quantum Internet, Scientific Reports, Nature, https://doi.org/10.1038/s41598-018-28801-3, https://www.nature.com/articles/s41598-018-28801-3 (2018).
Gyongyosi, L. & Imre, S. Entanglement-Gradient Routing for Quantum Networks, Scientific Reports, Nature, https://doi.org/10.1038/s41598-017-14394-w, https://www.nature.com/articles/s41598-017-14394-w (2017).
Gyongyosi, L. & Imre, S. A Survey on Quantum Computing Technology, Computer Science Review, Elsevier, https://doi.org/10.1016/j.cosrev.2018.11.002, ISSN: 1574-0137 (2018).
Rozpedek, F. et al. Optimizing practical entanglement distillation. Phys. Rev. A 97, 062333 (2018).
Humphreys, P. et al. Deterministic delivery of remote entanglement on a quantum network. Nature 558 (2018).
Liao, S.-K. et al. Satellite-to-ground quantum key distribution. Nature 549, 43–47 (2017).
Ren, J.-G. et al. Ground-to-satellite quantum teleportation. Nature 549, 70–73 (2017).
Hensen, B. et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature 526 (2015).
Hucul, D. et al. Modular entanglement of atomic qubits using photons and phonons. Nature Physics 11(1) (2015).
Noelleke, C. et al. Efficient Teleportation Between Remote Single-Atom Quantum Memories. Physical Review Letters 110, 140403 (2013).
Sangouard, N. et al. Quantum repeaters based on atomic ensembles and linear optics. Reviews of Modern Physics 83, 33 (2011).
Caleffi, M. End-to-End Entanglement Rate: Toward a Quantum Route Metric, 2017 IEEE Globecom, https://doi.org/10.1109/GLOCOMW.2017.8269080 (2018).
Caleffi, M. Optimal Routing for Quantum Networks. IEEE Access Vol 5, https://doi.org/10.1109/ACCESS.2017.2763325 (2017).
Caleffi, M., Cacciapuoti, A. S. & Bianchi, G. Quantum Internet: from Communication to Distributed Computing, arXiv:1805.04360 (2018).
Castelvecchi, D. The quantum internet has arrived, Nature, News and Comment, https://www.nature.com/articles/d41586-018-01835-3 (2018).
Cacciapuoti, A. S. et al. Quantum Internet: Networking Challenges in Distributed Quantum Computing, arXiv:1810.08421 (2018).
Chakraborty, K., Rozpedeky, F., Dahlbergz, A. & Wehner, S. Distributed Routing in a Quantum Internet, arXiv:1907.11630v1 (2019).
Khatri, S., Matyas, C. T., Siddiqui, A. U. & Dowling, J. P. Practical figures of merit and thresholds for entanglement distribution in quantum networks. Phys. Rev. Research 1, 023032 (2019).
Kozlowski, W. & Wehner, S. Towards Large-Scale Quantum Networks. Proc. of the Sixth Annual ACM International Conference on Nanoscale Computing and Communication, Dublin, Ireland, arXiv:1909.08396 (2019).
Pathumsoot, P. et al. Modeling of Measurement-based Quantum Network Coding on IBMQ Devices. arXiv:1910.00815v1 (2019).
Pal, S., Batra, P., Paterek, T. & Mahesh, T. S. Experimental localisation of quantum entanglement through monitored classical mediator. arXiv:1909.11030v1 (2019).
Briegel, H. J., Dur, W., Cirac, J. I. & Zoller, P. Quantum repeaters: the role of imperfect local operations in quantum communication. Phys. Rev. Lett. 81, 5932–5935 (1998).
Dur, W., Briegel, H. J., Cirac, J. I. & Zoller, P. Quantum repeaters based on entanglement purification. Phys. Rev. A 59, 169–181 (1999).
Van Loock, P. et al. Hybrid quantum repeater using bright coherent light. Phys. Rev. Lett. 96, 240501 (2006).
Simon, C. et al. Quantum Repeaters with Photon Pair Sources and Multimode Memories. Phys. Rev. Lett. 98, 190503 (2007).
Sangouard, N., Dubessy, R. & Simon, C. Quantum repeaters based on single trapped ions. Phys. Rev. A 79, 042340 (2009).
Gyongyosi, L. & Imre, S. Training Optimization for Gate-Model Quantum Neural Networks, Scientific Reports, Nature, https://doi.org/10.1038/s41598-019-48892-w (2019).
Gyongyosi, L. & Imre, S. Dense Quantum Measurement Theory, Scientific Reports, Nature, https://doi.org/10.1038/s41598-019-43250-2 (2019).
Gyongyosi, L. & Imre, S. State Stabilization for Gate-Model Quantum Computers, Quantum Information Processing, Springer Nature, https://doi.org/10.1007/s11128-019-2397-0 (2019).
Gyongyosi, L. & Imre, S. Quantum Circuit Design for Objective Function Maximization in Gate-Model Quantum Computers, Quantum Information Processing, https://doi.org/10.1007/s11128-019-2326-2 (2019).
Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018).
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, https://doi.org/10.1038/s41586-019-1666-5 (2019).
Harrow, A. W. & Montanaro, A. Quantum Computational Supremacy. Nature 549, 203–209 (2017).
Aaronson, S. & Chen, L. Complexity-theoretic foundations of quantum supremacy experiments. Proceedings of the 32nd Computational Complexity Conference, CCC ’17, pages 22:1–22:67 (2017).
Farhi, E., Goldstone, J., Gutmann, S. & Neven, H. Quantum Algorithms for Fixed Qubit Architectures. arXiv:1703.06199v1 (2017).
Farhi, E. & Neven, H. Classification with Quantum Neural Networks on Near Term Processors, arXiv:1802.06002v1 (2018).
Alexeev, Y. et al. Quantum Computer Systems for Scientific Discovery, arXiv:1912.07577 (2019).
Loncar, M. et al. Development of Quantum InterConnects for Next- Generation Information Technologies, arXiv:1912.06642 (2019).
Lloyd, S. & Weedbrook, C. Quantum generative adversarial learning. Phys. Rev. Lett. 121, arXiv:1804.09139 (2018).
Gisin, N. & Thew, R. Quantum Communication. Nature Photon 1, 165–171 (2007).
Xiao, Y. F. & Gong, Q. Optical microcavity: from fundamental physics to functional photonics devices. Science Bulletin 61, 185–186 (2016).
Zhang, W. et al. Quantum Secure Direct Communication with Quantum Memory. Phys. Rev. Lett. 118, 220501 (2017).
Enk, S. J., Cirac, J. I. & Zoller, P. Photonic channels for quantum communication. Science 279, 205–208 (1998).
Duan, L. M., Lukin, M. D., Cirac, J. I. & Zoller, P. Long-distance quantum communication with atomic ensembles and linear optics. Nature 414, 413–418 (2001).
Zhao, B., Chen, Z. B., Chen, Y. A., Schmiedmayer, J. & Pan, J. W. Robust creation of entanglement between remote memory qubits. Phys. Rev. Lett. 98, 240502 (2007).
Goebel, A. M. et al. Multistage Entanglement Swapping. Phys. Rev. Lett. 101, 080403 (2008).
Tittel, W. et al. Photon-echo quantum memory in solid state systems. Laser Photon. Rev 4, 244–267 (2009).
Dur, W. & Briegel, H. J. Entanglement purification and quantum error correction. Rep. Prog. Phys 70, 1381–1424 (2007).
Sheng, Y. B. & Zhou, L. Distributed secure quantum machine learning. Science Bulletin 62, 1025–1029 (2017).
Leung, D., Oppenheim, J. & Winter, A. IEEE Trans. Inf. Theory 56, 3478-90. (2010).
Kobayashi, H., Le Gall, F., Nishimura, H. & Rotteler, M. Perfect quantum network communication protocol based on classical network coding. Proceedings of 2010 IEEE International Symposium on Information Theory (ISIT) pp 2686–90 (2010).
Petz, D. Quantum Information Theory and Quantum Statistics 6 (Springer-Verlag, Heidelberg, Hiv, 2008).
Lloyd, S. Capacity of the noisy quantum channel. Physical Rev. A 55, 1613–1622 (1997).
Lloyd, S. The Universe as Quantum Computer. A Computable Universe: Understanding and exploring Nature as computation, Zenil, H. ed., World Scientific, Singapore, arXiv:1312.4455v1 (2013).
Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493–R2496 (1995).
Chou, C. et al. Functional quantum nodes for entanglement distribution over scalable quantum networks. Science 316(5829), 1316–1320 (2007).
Muralidharan, S., Kim, J., Lutkenhaus, N., Lukin, M. D. & Jiang, L. Ultrafast and Fault-Tolerant Quantum Communication across Long Distances. Phys. Rev. Lett. 112, 250501 (2014).
Yuan, Z. et al. Nature 454, 1098–1101 (2008).
Kobayashi, H., Le Gall, F., Nishimura, H. & Rotteler, M. General scheme for perfect quantum network coding with free classical communication. Lecture Notes in Computer Science (Automata, Languages and Programming SE-52 vol. 5555), Springer) pp 622–633 (2009).
Hayashi, M. Prior entanglement between senders enables perfect quantum network coding with modification. Physical Review A 76, 040301(R) (2007).
Hayashi, M., Iwama, K., Nishimura, H., Raymond, R. & Yamashita, S. Quantum network coding. Lecture Notes in Computer Science (STACS 2007 SE52 vol. 4393) ed. Thomas, W. & Weil, P. (Berlin Heidelberg: Springer) (2007).
Chen, L. & Hayashi, M. Multicopy and stochastic transformation of multipartite pure states. Physical Review A 83(2), 022331 (2011).
Schoute, E., Mancinska, L., Islam, T., Kerenidis, I. & Wehner, S. Shortcuts to quantum network routing. arXiv:1610.05238 (2016).
Distante, E. et al. Storing single photons emitted by a quantum memory on a highly excited Rydberg state. Nat. Commun. 8, 14072, https://doi.org/10.1038/ncomms14072 (2017).
Albrecht, B., Farrera, P., Heinze, G., Cristiani, M. & de Riedmatten, H. Controlled rephasing of single collective spin excitations in a cold atomic quantum memory. Phys. Rev. Lett. 115, 160501 (2015).
Choi, K. S. et al. Mapping photonic entanglement into and out of a quantum memory. Nature 452, 67–71 (2008).
Chaneliere, T. et al. Storage and retrieval of single photons transmitted between remote quantum memories. Nature 438, 833–836 (2005).
Fleischhauer, M. & Lukin, M. D. Quantum memory for photons: Dark-state polaritons. Phys. Rev. A 65, 022314 (2002).
Korber, M. et al. Decoherence-protected memory for a single-photon qubit. Nature Photonics 12, 18–21 (2018).
Yang, J. et al. Coherence preservation of a single neutral atom qubit transferred between magic-intensity optical traps. Phys. Rev. Lett. 117, 123201 (2016).
Ruster, T. et al. A long-lived Zeeman trapped-ion qubit. Appl. Phys. B 112, 254 (2016).
Neuzner, A. et al. Interference and dynamics of light from a distance-controlled atom pair in an optical cavity. Nat. Photon 10, 303–306 (2016).
Yang, S.-J., Wang, X.-J., Bao, X.-H. & Pan, J.-W. An efficient quantum light-matter interface with sub-second lifetime. Nat. Photon. 10, 381–384 (2016).
Uphoff, M., Brekenfeld, M., Rempe, G. & Ritter, S. An integrated quantum repeater at telecom wavelength with single atoms in optical fiber cavities. Appl. Phys. B 122, 46 (2016).
Zhong, M. et al. Optically addressable nuclear spins in a solid with a six-hour coherence time. Nature 517, 177–180 (2015).
Sprague, M. R. et al. Broadband single-photon-level memory in a hollow-core photonic crystal fibre. Nat. Photon 8, 287–291 (2014).
Gouraud, B., Maxein, D., Nicolas, A., Morin, O. & Laurat, J. Demonstration of a memory for tightly guided light in an optical nanofiber. Phys. Rev. Lett. 114, 180503 (2015).
Razavi, M., Piani, M. & Lutkenhaus, N. Quantum repeaters with imperfect memories: Cost and scalability. Phys. Rev. A 80, 032301 (2009).
Langer, C. et al. Long-lived qubit memory using atomic ions. Phys. Rev. Lett. 95, 060502 (2005).
Maurer, P. C. et al. Room-temperature quantum bit memory exceeding one second. Science 336, 1283–1286 (2012).
Steger, M. et al. Quantum information storage for over 180 s using donor spins in a 28Si semiconductor vacuum. Science 336, 1280–1283 (2012).
Bar-Gill, N., Pham, L. M., Jarmola, A., Budker, D. & Walsworth, R. L. Solid-state electronic spin coherence time approaching one second. Nature Commun 4, 1743 (2013).
Riedl, S. et al. Bose-Einstein condensate as a quantum memory for a photonic polarisation qubit. Phys. Rev. A 85, 022318 (2012).
Xu, Z. et al. Long lifetime and high-fidelity quantum memory of photonic polarisation qubit by lifting Zeeman degeneracy. Phys. Rev. Lett. 111, 240503 (2013).
Zhu, F., Zhang, W., Sheng, Y. B. & Huang, Y. D. Experimental long-distance quantum secret direct communication. Sci. Bull 62, 1519 (2017).
Wu, F. Z. et al. High-capacity quantum secure direct communication with two-photon six-qubit hyperentangled states. Sci. China Phys. Mech. Astron. 60, 120313 (2017).
Chen, S. S., Zhou, L., Zhong, W. & Sheng, Y. B. Three-step three-party quantum secure direct communication. Sci. China Phys. Mech. Astron 61, 090312 (2018).
Niu, P. H. et al. Measurement-device-independent quantum communication without encryption. Sci. Bull. 63, 1345–1350 (2018).
Biamonte, J. et al. Quantum Machine Learning. Nature 549, 195–202 (2017).
Lloyd, S., Mohseni, M. & Rebentrost, P. Quantum algorithms for supervised and unsupervised machine learning. arXiv:1307.0411 (2013).
Lloyd, S., Mohseni, M. & Rebentrost, P. Quantum principal component analysis. Nature Physics 10, 631 (2014).
Kok, P. et al. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys. 79, 135–174 (2007).
Bacsardi, L. On the Way to Quantum-Based Satellite Communication. IEEE Comm. Mag. 51(08), 50–55 (2013).
Gyongyosi, L., Imre, S. & Nguyen, H. V. A Survey on Quantum Channel Capacities. IEEE Communications Surveys and Tutorials, https://doi.org/10.1109/COMST.2017.2786748 (2018).
Gyongyosi, L., Bacsardi, L. & Imre, S. A Survey on Quantum Key Distribution, Infocom. J XI, 2, pp. 14–21 (2019).
Imre, S. & Gyongyosi, L. Advanced Quantum Communications - An Engineering Approach. (Wiley-IEEE Press, New Jersey, 2013).
Chien, J.-T. Source Separation and Machine Learning, Academic Press (2019).
Yang, P.-K., Hsu, C.-C. & Chien, J.-T., Bayesian factorization and selection for speech and music separation. In: Proc. of Annual Conference of International Speech Communication Association, pp. 998–1002 (2014).
Yang, P.-K., Hsu, C.-C. & Chien, J.-T., Bayesian singing-voice separation. In: Proc. of Annual Conference of International Society for Music Information Retrieval (ISMIR), pp. 507–512 (2014).
Chien, J.-T. & Yang, P.-K. Bayesian factorization and learning for monaural source separation. IEEE/ACM Transactions on Audio, Speech and Language Processing 24(1), 185–195 (2016).
Bishop, C. M. Pattern Recognition and Machine Learning. Springer Science (2006).
Vembu, S. & Baumann, S. Separation of vocals from polyphonic audio recordings. In: Proc. of ISMIR, pages 375–378 (2005).
Lee, D. D. & Seung, H. S. Algorithms for nonnegative matrix factorization. Advances in Neural Information Processing Systems, 556–562 (2000).
Cemgil, A. T. Bayesian inference for nonnegative matrix factorisation models. Computational Intelligence and Neuroscience, 785152 (2009).
Schmidt, M. N., Winther, O. & Hansen, L. K. Bayesian non-negative matrix factorization. In: Proc. of ICA, 540–547 (2009).
Tibshirani, R. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B 58(1), 267–288 (1996).
Brown, J. C. Calculation of a Constant Q spectral transform. Journal of the Acoustic Society of America 89(1), 425–434 (1991).
Quatieri, T. F. Discrete-Time Speech Signal Processing: Principles and Practice, Prentice Hall, ISBN-10: 013242942X, ISBN-13: 978-0132429429 (2002).
Jaiswal, R. et al. Clustering NMF Basis Functions Using Shifted NMF for Monaural Sound Source Separation. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2011).
FitzGerald, D., Cranitch, M. & Coyle, E. Shifted Nonnegative matrix factorisation for sound source separation. IEEE Workshop of Statistical Signal Processing, Bordeaux, France (2005).
Bader, B. W. & Kolda, T. G. MATLAB Tensor Classes for Fast Algorithm Prototyping, Sandia National Laboratories Report, SAND2004-5187 (2004).
Sherrill, C. D. A Brief Review of Elementaary Quantum Chemistry, Lecture Notes, web: http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html (2001).
Acknowledgements
The research reported in this paper has been supported by the National Research, Development and Innovation Fund (TUDFO/51757/2019-ITM, Thematic Excellence Program). This work was partially supported by the National Research Development and Innovation Office of Hungary (Project No. 2017-1.2.1-NKP-2017-00001), by the Hungarian Scientific Research Fund - OTKA K-112125 and in part by the BME Artificial Intelligence FIKP grant of EMMI (BME FIKP-MI/SC).
Author information
Authors and Affiliations
Contributions
L.GY. designed the protocol and wrote the manuscript. L.GY. and S.I. analyzed the results. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gyongyosi, L., Imre, S. Optimizing High-Efficiency Quantum Memory with Quantum Machine Learning for Near-Term Quantum Devices. Sci Rep 10, 135 (2020). https://doi.org/10.1038/s41598-019-56689-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598-019-56689-0