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arXiv:0711.0721 · Journal of Mathematical Physics

Relations between convergence rates in Schatten p-norms

Paolo Albini, Alessandro Toigo, Veronica Umanità

Published Jan 1, 2008• 0 citations
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Abstract

In quantum estimation theory and quantum tomography, the quantum state obtained by sampling converges to the “true” unknown density matrix under topologies that are different from the natural notion of distance in the space of quantum states, i.e., the trace class norm. In this paper, we address such problem, finding relations between the rates of convergence in the Schatten p-norms and in the trace class norm.

DOI
10.1063/1.2830521

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In quantum estimation theory and quantum tomography, the quantum state obtained by sampling converges to the “true” unknown density matrix under topologies that are different from the natural notion of distance in the space of quantum states, i.e., the trace class norm. In this paper, we address such problem, finding relations between the rates of convergence in the Schatten p-norms and in the trace class norm.
Let Sm\mathcal{S}_{m}Sm​ be the set of all m×mm\times mm×m density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix ρ∈Sm\rho\in\mathcal{S}_{m}ρ∈Sm​ based on outcomes of nnn measurements of observables X1,…,Xn∈HmX_{1},\dots,X_{n}\in\mathbb{H}_{m}X1​,…,Xn​∈Hm​ (Hm\mathbb{H}_{m}Hm​ being the space of m×mm\times mm×m Hermitian matrices) for a quantum system identically prepared nnn times in state ρ.\rho.ρ. Outcomes Y1,…,YnY_{1},\dots,Y_{n}Y1​,…,Yn​ of such measurements could be described by a trace regression model in which Eρ(Yj∣Xj)=tr(ρXj),j=1,…,n.\mathbb{E}_{\rho}(Y_{j}|X_{j})={\rm tr}(\rho X_{j}),j=1,\dots,n.Eρ​(Yj​∣Xj​)=tr(ρXj​),j=1,…,n. The design variables X1,…,XnX_{1},\dots,X_{n}X1​,…,Xn​ are often sampled at random from the uniform distribution in an orthonormal basis {E1,…,Em2}\{E_{1},\dots,E_{m^{2}}\}{E1​,…,Em2​} of Hm\mathbb{H}_{m}Hm​ (such as Pauli basis). The goal is to estimate the unknown density matrix ρ\rhoρ based on the data (X1,Y1),…,(Xn,Yn).(X_{1},Y_{1}),\dots,(X_{n},Y_{n}).(X1​,Y1​),…,(Xn​,Yn​). Let Z^:=m2n∑j=1nYjXj\hat{Z}:=\frac{m^{2}}{n}\sum_{j=1}^{n}Y_{j}X_{j}Z^:=nm2​∑j=1n​Yj​Xj​ and let ρˇ\check{\rho}ρˇ​ be the projection of Z^\hat{Z}Z^ onto the convex set Sm\mathcal{S}_{m}Sm​ of density matrices. It is shown that for estimator ρˇ\check{\rho}ρˇ​ the minimax lower bounds in classes of low rank density matrices (established earlier) are attained up logarithmic factors for all Schatten ppp-norm distances, p∈[1,∞]p\in[1,\infty]p∈[1,∞] and for Bures version of quantum Hellinger distance. Moreover, for a slightly modified version of estimator ρˇ\check{\rho}ρˇ​ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.
Density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. Many quantum systems of physical interest can be represented as high-dimensional low rank density matrices. A popular problem in quantum state tomography (QST) is to estimate the unknown low rank density matrix of a quantum system by conducting Pauli measurements. Our main contribution is twofold. First, we establish the minimax lower bounds in Schatten ppp-norms with 1≤p≤+∞1\leq p\leq+\infty1≤p≤+∞ for low rank density matrices estimation by Pauli measurements. In our previous paper [14], these minimax lower bounds are proved under the trace regression model with Gaussian noise and the noise is assumed to have common variance. In this paper, we prove these bounds under the Binomial observation model which meets the actual model in QST. Second, we study the Dantzig estimator (DE) for estimating the unknown low rank density matrix under the Binomial observation model by using Pauli measurements. In our previous papers [14] and [25], we studied the least squares estimator and the projection estimator, where we proved the optimal convergence rates for the least squares estimator in Schatten ppp-norms with 1≤p≤21\leq p\leq21≤p≤2 and, under a stronger condition, the optimal convergence rates for the projection estimator in Schatten ppp-norms with 1≤p≤+∞1\leq p\leq+\infty1≤p≤+∞. In this paper, we show that the results of these two distinct estimators can be simultaneously obtained by the Dantzig estimator. Moreover, better convergence rates in Schatten norm distances can be proved for Dantzig estimator under conditions weaker than those needed in [14] and [25]. When the objective function of DE is replaced by the negative von Neumann entropy, we obtain sharp convergence rate in Kullback-Leibler divergence.
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, in the case of Pauli measurements) with explicit dependence of the bounds on the rank and other complexity parameters. Such bounds are established for several statistically relevant distances, including quantum versions of Kullback-Leibler divergence (relative entropy distance) and of Hellinger distance (so called Bures distance), and Schatten ppp-norm distances. Sharp upper bounds and oracle inequalities for least squares estimator with von Neumann entropy penalization are obtained showing that minimax lower bounds are attained (up to logarithmic factors) for these distances.
Efficient measures to determine the similarity of quantum states, such as the fidelity metric, have been widely studied. In this paper, we address the problem of defining a similarity measure for quantum operations that can be efficiently estimated. Given two quantum operations, U1{U}_{1}U1​ and U2{U}_{2}U2​, represented in their circuit forms, we first develop a quantum sampling circuit to estimate the normalized Schatten 2-norm of their difference ($\ensuremath{\parallel}{U}_{1}\ensuremath{-}{U}_{2}{\ensuremath{\parallel}}_{{S}_{2}}$) with precision $\ensuremath{\epsilon}$, using only one clean qubit and one classical random variable. We prove a $\text{Poly}(\frac{1}{\ensuremath{\epsilon}})$ upper bound on the sample complexity, which is independent of the size of the quantum system. We then show that such a similarity metric is directly related to a functional definition of similarity of unitary operations using the conventional fidelity metric of quantum states (F\mathcal{F}F): If $\ensuremath{\parallel}{U}_{1}\ensuremath{-}{U}_{2}{\ensuremath{\parallel}}_{{S}_{2}}$ is sufficiently small (e.g., $\ensuremath{\le}\frac{\ensuremath{\epsilon}}{1+\sqrt{2(1/\ensuremath{\delta}\ensuremath{-}1)}}$) then the fidelity of states obtained by processing the same randomly and uniformly picked pure state $|\ensuremath{\psi}\ensuremath{\rangle}$ is as high as needed $[\mathcal{F}({U}_{1}|\ensuremath{\psi}\ensuremath{\rangle},{U}_{2}|\ensuremath{\psi}\ensuremath{\rangle})\ensuremath{\ge}1\ensuremath{-}\ensuremath{\epsilon}]$ with probability exceeding $1\ensuremath{-}\ensuremath{\delta}$. We provide example applications of this efficient similarity metric estimation framework to quantum circuit learning tasks, such as finding the square root of a given unitary operation.
We consider the quantum complexity of computing Schatten p-norms and related quantities, and find that the problem of estimating these quantities is closely related to the one clean qubit model of computation. We show that the problem of approximating Tr(|A|^p) for a log-local n-qubit Hamiltonian A and p=poly(n), up to a suitable level of accuracy, is contained in DQC1; and that approximating this quantity up to a somewhat higher level of accuracy is DQC1-hard. In some cases the level of accuracy achieved by the quantum algorithm is substantially better than a natural classical algorithm for the problem. The same problem can be solved for arbitrary sparse matrices in BQP. One application of the algorithm is the approximate computation of the energy of a graph.
We consider the quantum complexity of computing Schatten p-norms and related quantities, and find that the problem of estimating these quantities is closely related to the one clean qubit model of computation. We show that the problem of approximating Tr(|A|^p) for a log-local n-qubit Hamiltonian A and p=poly(n), up to a suitable level of accuracy, is contained in DQC1; and that approximating this quantity up to a somewhat higher level of accuracy is DQC1-hard. In some cases the level of accuracy achieved by the quantum algorithm is substantially better than a natural classical algorithm for the problem. The same problem can be solved for arbitrary sparse matrices in BQP. One application of the algorithm is the approximate computation of the energy of a graph.