Many applications of quantum simulation require to prepare and then characterize quantum states by performing an efficient partial tomography to estimate observables corresponding to $k$-body reduced density matrices ($k$-RDMs). For instance, variational algorithms for the quantum simulation of chemistry usually require that one measure the fermionic 2-RDM. While such marginals provide a tractable description of quantum states from which many important properties can be computed, their determination often requires a prohibitively large number of circuit repetitions. Here we describe a method by which all elements of $k$-body qubit RDMs acting on $N$ qubits can be directly measured with a number of circuits scaling as ${\cal O}(3^{k} \log^{k-1}\! N)$, an exponential improvement in $N$ over prior art. Next, we show that if one is able to implement a linear depth circuit on a linear array prior to measurement, then one can directly measure all elements of the fermionic 2-RDM using only ${\cal O}(N^2)$ circuits. We prove that this result is asymptotically optimal, thus establishing an exponential separation between the number of circuits required to directly measure all elements of qubit versus fermion RDMs. We further demonstrate a technique to estimate the expectation value of any linear combination of fermionic 2-RDM elements using ${\cal O}(N^4 / ω)$ circuits, each with only ${\cal O}(ω)$ gates on a linear array where $ω\leq N$ is a free parameter. We expect these results will improve the viability of many proposals for near-term quantum simulation.
The success of quantum information processing applications relies on accurate and efficient characterization of quantum states, especially nearly-pure states. In this work, we investigate a procedure for adaptive qubit state tomography which achieves $O(1/N)$ scaling in accuracy for large $N$. We analyze the performance of the adaptive protocol on the characterization of pure, mixed and nearly-pure states, and clarify the interplay between the purity of the state and the statistical fluctuations of the measurement results. Our results highlight the subtle difference between the characterizatio…
We show that quantum state tomography with perfect knowledge of the measurement apparatus proves to be, in some instances, inferior to strategies discarding all information about the measurement at hand, as in the case of data pattern tomography. In those scenarios, the larger uncertainty about the measurement is traded for the smaller uncertainty about the reconstructed signal. This effect is more pronounced for minimal or nearly minimal informationally complete measurement settings, which are of utmost practical importance.
We discuss quantum state tomography via a stepwise reconstruction of the eigenstates of the mixed states produced in experiments. Our method is tailored to the experimentally relevant class of nearly pure states or simple mixed states, which exhibit dominant eigenstates and thus lend themselves to low-rank approximations. The developed scheme is applicable to any pure-state tomography method, promoting it to mixed-state tomography. Here, we demonstrate it with machine learning-inspired pure-state tomography based on neural-network representations of quantum states. The latter have been shown t…
Quantum process tomography is a procedure by which the unknown dynamical evolution of an open quantum system can be fully experimentally characterized. We demonstrate explicitly how this procedure can be implemented with a nuclear magnetic resonance quantum computer. This allows us to measure the fidelity of a controlled-not logic gate and to experimentally investigate the error model for our computer. Based on the latter analysis, we test an important assumption underlying nearly all models of quantum error correction, the independence of errors on different qubits.
The need to perform quantum state tomography on ever larger systems has spurred a search for methods that yield good estimates from incomplete data. We study the performance of compressed sensing (CS) and least squares (LS) estimators in a fast protocol based on continuous measurement on an ensemble of cesium atomic spins. Both efficiently reconstruct nearly pure states in the 16-dimensional ground manifold, reaching average fidelities FCS = 0.92 and FLS = 0.88 using similar amounts of incomplete data. Surprisingly, the main advantage of CS in our protocol is an increased robustness to experim…
We introduce the concept of selective quantum state tomography or SQST, a tomographic scheme that enables a user to estimate arbitrary elements of an unknown quantum state using a fixed measurement record. We demonstrate how this may be done with the following notable advantages (i) a number of state copies that depends only on the desired precision of the estimation, rather than the dimension of the unknown state; (ii) a similar reduction in the requisite classical memory and computational cost; (iii) an approach to state tomography using $O(ε^{-2}\log d)$ state copies for maximum norm error …
We present the experimental reconstruction of the Wigner function of an individual electronic spin qubit associated with a nitrogen-vacancy (NV) center in diamond at room temperature. This spherical Wigner function contains the same information as the density matrix for arbitrary spin systems. As an example, we exactly characterize the quantum state of a single qubit undergoing a nearly pure dephasing process by Wigner function. The fidelities and purities during this process are extracted from the experimental reconstructed Wigner functions, whose dynamics agree with the theoretical predictio…
We report an experimental realization of adaptive Bayesian quantum state tomography for two-qubit states. Our implementation is based on the adaptive experimental design strategy proposed in [F.Huszár and N.M.T.Houlsby, Phys.Rev.A 85, 052120 (2012)] and provides an optimal measurement approach in terms of the information gain. We address the practical questions, which one faces in any experimental application: the influence of technical noise, and behavior of the tomographic algorithm for an easy to implement class of factorized measurements. In an experiment with polarization states of entang…
Quantum tomography is a critically important tool to evaluate quantum hardware, making it essential to develop optimized measurement strategies that are both accurate and efficient. We compare a variety of strategies using nearly pure test states. Those that are informationally complete for all states are found to be accurate and reliable even in the presence of errors in the measurements themselves, while those designed to be complete only for pure states are far more efficient but highly sensitive to such errors. Our results highlight the unavoidable tradeoffs inherent to quantum tomography.
Identifiability and Estimation Precision in Quantum Network Tomography with Imperfect Bell-State Measurements
CONNECT Research Centre, School of Engineering, Trinity College Dublin（都柏林三一学院工程学院连接研究中心）；Manning College of Information and Computer Science, University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校曼宁信息与计算机科学学院）；Department of Computing, School of Science, South East Technological University（东南理工大学理学院计算系）
The recovery of an unknown density matrix of large size requires huge computational resources. The recent Factored Gradient Descent (FGD) algorithm and its variants achieved state-of-the-art performance since they could mitigate the dimensionality barrier by utilizing some of the underlying structures of the density matrix. Despite their theoretical guarantee of a linear convergence rate, the convergence in practical scenarios is still slow because the contracting factor of the FGD algorithms depends on the condition number $κ$ of the ground truth state. Consequently, the total number of itera…
We consider the statistical problem of `compressive' estimation of low rank states with random basis measurements, where the estimation error is expressed terms of two metrics - the Frobenius norm and quantum infidelity. It is known that unlike the case of general full state tomography, low rank states can be identified from a reduced number of observables' expectations. Here we investigate whether for a fixed sample size $N$, the estimation error associated to a `compressive' measurement setup is `close' to that of the setting where a large number of bases are measured. In terms of the Froben…
Single-acquisition tomography of photonic qubits with structured media
University of Ottawa（渥太华大学）；Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）；National Research Council of Canada（加拿大国家研究委员会）；McGill University（麦吉尔大学）；Chapman University（查普曼大学）
Characterizing Photonic Partial Distinguishability Through Gram Matrix Tomography
Center for Quantum Enabled-Computing, Center for Theoretical Physics of the Polish Academy of Sciences（波兰科学院理论物理中心量子使能计算中心）；International Iberian Nanotechnology Laboratory (INL)（伊比利亚纳米技术国际实验室）；Centre for Quantum Information and Communication, École polytechnique de Bruxelles, CP 165/59, Université libre de Bruxelles（布鲁塞尔自由大学布鲁塞尔理工学院量子信息与通信中心）；Instituto de Física, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学物理研究所）
It is now experimentally possible to entangle thousands of qubits, and efficiently measure each qubit in parallel in a distinct basis. To fully characterize an unknown entangled state of $n$ qubits, one requires an exponential number of measurements in $n$, which is experimentally unfeasible even for modest system sizes. By leveraging (i) that single-qubit measurements can be made in parallel, and (ii) the theory of perfect hash families, we show that all $k$-qubit reduced density matrices of an $n$ qubit state can be determined with at most $e^{\mathcal{O}(k)} \log^2(n)$ rounds of parallel me…
Following [OW16], we continue our analysis of: (1) "Quantum tomography", i.e., learning a quantum state, i.e., the quantum generalization of learning a discrete probability distribution; (2) The distribution of Young diagrams output by the RSK algorithm on random words. Regarding (2), we introduce two powerful new tools: (i) A precise upper bound on the expected length of the longest union of $k$ disjoint increasing subsequences in a random length-$n$ word with letter distribution $α_1 \geq α_2 \geq \cdots \geq α_d$; (ii) A new majorization property of the RSK algorithm that allows one to anal…
We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras. Subject to some reasonable constraints, we show that no such composite exists having the exceptional Jordan algebra as a direct summand. We then construct several dagger compact categories of such Jordan-algebraic models. One of these neatly unifies real, complex and quaternionic mixed-state quantum mechanics, with the exception of the quaternionic "bit". Another is similar, except in that (i) it excludes the quaternionic bit, and (ii) the composite of two complex quantum systems comes wit…
The resources required to characterise the dynamics of engineered quantum systems-such as quantum computers and quantum sensors-grow exponentially with system size. Here we adapt techniques from compressive sensing to exponentially reduce the experimental configurations required for quantum process tomography. Our method is applicable to dynamical processes that are known to be nearly-sparse in a certain basis and it can be implemented using only single-body preparations and measurements. We perform efficient, high-fidelity estimation of process matrices on an experiment attempting to implemen…
Lightwave periodic driving of nearly dissipation-less currents has recently emerged as a universal control concept for both superconducting (SC) and topological electronics applications. While exciting progress has been made towards THz-driven superconductivity, our understanding of the interactions able to drive non-equilibrium pairing is still limited, partially due to the lack of direct measurements of high-order correlation functions. Such measurements would exceed conventional single-particle spectroscopies and perturbative responses to fully characterize quantum states far-from-equilibri…
Quantum entanglement of mechanical systems emerges when distinct objects move with such a high degree of correlation that they can no longer be described separately. Although quantum mechanics presumably applies to objects of all sizes, directly observing entanglement becomes challenging as masses increase, requiring measurement and control with a vanishingly small error. Here, using pulsed electromechanics, we deterministically entangle two mechanical drumheads with masses of 70 pg. Through nearly quantum-limited measurements of the position and momentum quadratures of both drums, we perform …
We propose a protocol to improve the accuracy of direct complex state measurements (DSM) by using rebits in real Hilbert spaces. We show that to improve the accuracy, the initial complex state should be decomposed into the real and imaginary parts and stored in an extended state (rebit) which can be tracked individually by two bases of an extra qubit. For pure states, the numerical calculations show that the trace distances between the true state and the reconstructed state obtained from the rebit method are more precise than those ones obtained from the usual DSM and quantum state tomography …
