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International Journal of Quantum Information

QUANTUM SYSTEM IDENTIFICATION

Li‐Yi Hsu

Published Nov 1, 2005• 1 citation
Publisher

Abstract

The joint measurement approach has been shown to detect more information in quantum system. In this paper, we propose a quantum identification algorithm based on the joint quantum measurement. Through this, we can collectively compare an unknown quantum composite system with those in the database. In addition, this study will show that the proposed quantum algorithm can speed up the identification process quadratically.

DOI
10.1142/s0219749905001390

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The joint measurement approach has been shown to detect more information in quantum system. In this paper, we propose a quantum identification algorithm based on the joint quantum measurement. Through this, we can collectively compare an unknown quantum composite system with those in the database. In addition, this study will show that the proposed quantum algorithm can speed up the identification process quadratically.
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Precise identification of parameters governing quantum processes is a critical task for quantum information and communication technologies. In this Letter, we consider a setting where system evolution is determined by a parametrized Hamiltonian, and the task is to estimate these parameters from temporal records of a restricted set of system observables (time traces). Based on the notion of system realization from linear systems theory, we develop a constructive algorithm that provides estimates of the unknown parameters directly from these time traces. We illustrate the algorithm and its robustness to measurement noise by applying it to a one-dimensional spin chain model with variable couplings.
System identification and inversion are two closely related problems, which lead to various configurations. In the quantum framework, system identification was only studied in the non-blind (or supervised) mode, i.e. with known input states (in addition to known output states). It is then called quantum process tomography. We here develop its blind (or unsupervised) version, operating with unknown (unentangled) input states (which are here not controlled by an automated loop). Our approach takes advantage of the blind quantum source separation (i.e. blind multi-qubit system inversion) methods based on disentanglement that we recently introduced. It is detailed for Heisenberg spin coupling and could be extended to other classes of processes. It estimates the parameter values of the Heisenberg Hamiltonian.
The aim of quantum system identification is to estimate the ingredients inside a black box, in which some quantum-mechanical unitary process takes place, by just looking at its input-output behavior. Here we establish a basic and general framework for quantum system identification, that allows us to classify how much knowledge about the quantum system is attainable, in principle, from a given experimental setup. We show that controllable closed quantum systems can be estimated up to unitary conjugation. Prior knowledge on some elements of the black box helps the system identification. We present an example in which a Bell measurement is more efficient to identify the system. When the topology of the system is known, the framework enables us to establish a general criterion for the estimability of the coupling constants in its Hamiltonian.
Many methods for processing classical or quantum data are based on estimating some parameters, e.g. those of adaptive filters, artificial neural networks (including deep learning approaches), blind source separation systems or quantum gates. For classical signals, these methods include stochastic algorithms, such as stochastic gradient descent. We here first introduce a partly related, general, type of stochastic algorithms for quantum data and we prove the asymptotic efficiency of the proposed estimator. We then show the attractiveness of this stochastic approach for the quantum version of blind multiple-input multiple-output system identification and blind source separation: the resulting model estimation methods can operate with a single copy of each considered quantum state, whereas the previous methods require many copies of the same (unknown) states to be available and are thus "less blind". Numerical tests show that good performance is obtained with 104such quantum states.
We introduce a novel quantum algorithm for similar subgraph identification in form of an NP-hard cardinality-constrained binary quadratic optimization problem. Given a weighted reference graph with Laplacian B\boldsymbol{B}B, our algorithm determines the subgraph featuring Laplacian B′\boldsymbol{B'}B′ on the same vertex set, but xxx out of NNN inactive edges, minimizing the Frobenius distance ∣∣B−B′∣∣F2||\boldsymbol{B} - \boldsymbol{B'}||_\mathrm{F}^2∣∣B−B′∣∣F2​. We represent the (Nx)\binom{N}{x}(xN​) graph topologies by an equal-weight superposition in form of a Dicke state, enabling controlled transformations applied to the quantum state associated with the vectorized Laplacian of the reference graph. Combined with amplitude estimation and a minimum finding approach, our algorithm provides a polynomial speed up O(Nx/x!Nlog⁡log⁡N)\mathcal{O}(\sqrt{N^{x}/x!}N\log\log N)O(Nx/x!​NloglogN) compared to O(Nx+1/x!)\mathcal{O}(N^{x+1}/x!)O(Nx+1/x!) of classical brute-force search algorithms. We demonstrate the application of our method on standard test cases, which represent electric power grids, by reconstructing ∣∣B−B′∣∣F2||\boldsymbol{B} -\boldsymbol{B'}||_\mathrm{F}^2∣∣B−B′∣∣F2​ from measurements and show how our approach can be additionally used to calculate energy functional like quadratic forms of the Laplacians with respect to a given vector.