We consider the problem of jointly estimating expectation values of many Pauli observables, a crucial subroutine in variational quantum algorithms. Starting with randomized measurements, we propose an efficient derandomization procedure that iteratively replaces random single-qubit measurements with fixed Pauli measurements; the resulting deterministic measurement procedure is guaranteed to perform at least as well as the randomized one. In particular, for estimating any $L$ low-weight Pauli observables, a deterministic measurement on only of order $\log(L)$ copies of a quantum state suffices. In some cases, for example when some of the Pauli observables have a high weight, the derandomized procedure is substantially better than the randomized one. Specifically, numerical experiments highlight the advantages of our derandomized protocol over various previous methods for estimating the ground-state energies of small molecules.
A quantum observable and a channel are considered compatible if they form parts of the same measurement device, otherwise they are incompatible. Constrains on compatibility between observables and channels can be quantified via relations highlighting the necessary trade-offs between noise and disturbance within quantum measurements. In this paper we shall discuss the general properties of these compatibility relations, and then fully characterize the compatibility conditions for an unbiased qubit observable and a Pauli channel. The implications of the characterization are demonstrated on some …
We obtain uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Rényi entropies of order $α\in(0;1]$. It is shown that these entropic bounds are tight in the sense that they are always reached with certain pure states. A new result is the conditions for equality in Rényi-entropy uncertainty relations for the Pauli observables. Upper entropic bounds in the pure-state case are also novel. Combining the presented bounds leads to a band, in which the rescaled average Rényi $α$-entropy ranges for a pure measured state. A width of this band is …
It is known that nonadditive quantum codes are more optimal for error correction when compared to stabilizer codes. The class of codeword stabilized codes (CWS) provides tools to obtain new nonadditive quantum codes by reducing the problem to finding nonlinear classical codes. In this work, we establish some results on the kind of non-Pauli operators that can be used as decoding observables for CWS codes and describe a procedure to obtain these observables.
Sequential weak measurements of non-commuting observables is not only fundamentally interesting in quantum measurement but also shown potential in various applications. The previous reported methods, however, can only realize limited sequential weak measurements experimentally. In this Letter, we propose the realization of sequential measurements of arbitrary observables and experimentally demonstrate for the first time the measurement of sequential weak values of three non-commuting Pauli observables by using genuine single photons. The results presented here will not only improve our underst…
The definition due to Accardi of a pair of complementary observables is adapted to the context of the Lie algebra $ su(2) $. We show that the pair of Pauli matrices $ A,B $ associated to the unit directions $ α$ and $ β$ in $ \mathbb{R}^{3} $ are Accardi complementary if and only if $ α$ and $ β$ are orthogonal if and only if $ A $ and $ B $ are orthogonal. In particular, any pair of the standard triple of Pauli matrices is complementary.
We present self-testing protocols to certify the presence of tensor products of Pauli measurements on maximally entangled states of local dimension $2^n$ for $n\in\mathbb{N}$. This provides self-tests of sets of informationally complete measurements in arbitrarily high dimension. We then show that this can be used for the device-independent certification of the entanglement of all bipartite entangled states by exploiting a connection to measurement device-independent entanglement witnesses and quantum networks. This work extends a more compact parallel work on the same subject and provides all…
Computing noise-canceling observables via Pauli propagation
IBM Quantum（IBM量子）；IBM Research – Silicon Valley（IBM硅谷研究院）；IBM T.J. Watson Research Center（IBM T.J.沃森研究中心）；IBM Research Europe – Zurich（IBM欧洲苏黎世研究院）；Research Triangle Park（研究三角园区）；IBM Research Cambridge（IBM剑桥研究院）
We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the N-qubit Pauli group for N >= 4, thus adding to the proofs that have been presented earlier for the two- and three-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large numbe…
We show that there exist informationally incomplete phase space observables such that the Cartesian margins are informationally equivalent with position and momentum. This shows that it is possible to reconstruct the position and momentum distributions of a quantum system from the statistics of a single observable, and thus a single measurement, even though the state of the system is not uniquely determined by the statistics.
The Pauli operators (tensor products of Pauli matrices) provide a complete basis of operators on the Hilbert space of N qubits. We prove that the set of 4^N-1 Pauli operators may be partitioned into 2^N+1 distinct subsets, each consisting of 2^N-1 internally commuting observables. Furthermore, each such partitioning defines a unique choice of 2^N+1 mutually unbiased basis sets in the N-qubit Hilbert space. Examples for 2 and 3 qubit systems are discussed with emphasis on the nature and amount of entanglement that occurs within these basis sets.
Reducing the number of measurements required to estimate the expectation value of an observable is crucial for the variational quantum eigensolver to become competitive with state-of-the-art classical algorithms. To measure complicated observables such as a molecular electronic Hamiltonian, one of the common strategies is to partition the observable into linear combinations (fragments) of mutually commutative Pauli products. The total number of measurements for obtaining the expectation value is then proportional to the sum of variances of individual fragments. We propose a method that lowers …
We address measurement schemes where certain observables are chosen at random within a set of non-degenerate isospectral observables and then measured on repeated preparations of a physical system. Each observable has a given probability to be measured, and the statistics of this generalized measurement is described by a positive operator-valued measure (POVM). This kind of schemes are referred to as quantum roulettes since each observable is chosen at random, e.g. according to the fluctuating value of an external parameter. Here we focus on quantum roulettes for qubits involving the measureme…
We extend the concept of Anderson localization, the confinement of quantum information in a spatially irregular potential, to quantum circuits. Considering matchgate circuits, generated by time-dependent spin-1/2 XY Hamiltonians, we give an analytic formula for the out-of-time-ordered correlator of a local observable, and show that it can be efficiently evaluated by a classical computer even when the explicit Heisenberg time evolution cannot. Because this quantity bounds the average error incurred by truncating the evolution to a spatially limited region, we demonstrate dynamical localization …
In most Bell tests, the measurement settings are specially chosen so that the maximal quantum violations of the Bell inequalities can be detected, or at least, the violations are strong enough to be observed. Such choices can usually associate the corresponding Bell operators to a kind of effective observables, called pseudo Pauli operators, providing us a more intuitive understanding of Bell nonlocality in some sense. Based on that, a more general quantum-to-classical approach for the constructions of Bell inequalities is developed. Using this approach, one can not only derive several kinds o…
The goal of the paper is to check whether the real eigenstates of the observables in the single qudit Pauli group may lead to quantum contextuality, the property that mutually compatible and independent experiments depend on each other. We find that quantum contextuality crops up at dimension twelve in various configurations with a few rays. We use the Shannon capacity for characterizing the corresponding orthogonality graphs. Some arithmetical properties underlying the qudit contextuality are outlined.
We present a noise deconvolution technique for obtaining noiseless expectation values of noisy observables at the output of multiqubit quantum channels. For any number of qubits or in the presence of correlations, our protocol applies to any mathematically invertible noise model, even when its inverse map is not physically implementable, i.e. when it is neither completely-positive nor trace-preserving. For a generic observable affected by Pauli noise it provides a quadratic speedup, always producing a rescaling of its Pauli basis components. We show that it is still possible to achieve the dec…
We introduce a new generalization of the Pauli channels using the mutually unbiased measurement operators. The resulting channels are bistochastic but their eigenvectors are not unitary. We analyze the channel properties, such as complete positivity, entanglement breaking, and multiplicativity of maximal output purity. We illustrate our results with the maps constructed from the Gell-Mann matrices and the Heisenberg-Weyl observables.
Pauli Weight Hamiltonian Term Selection for Optimized Machine Learning Based Quantum Error Mitigation
Bandung Institute of Technology（万隆理工学院）；National Research and Innovation Agency (BRIN)（国家研究与创新署）；National University of Singapore（新加坡国立大学）；Nanyang Technological University（南洋理工大学）
Enhanced measurements on quantum computers via the simultaneous probing of non-commuting Pauli operators
University of Exeter（埃克塞特大学）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Department of Combinatorics & Optimization, University of Waterloo（滑铁卢大学组合优化系）
Derandomized shallow shadows: Efficient Pauli learning with bounded-depth circuits
Harvard University（哈佛大学）；Harvard-Smithsonian Center for Astrophysics（哈佛-史密森尼天体物理中心）；University of Maryland（马里兰大学）；Joint Quantum Institute（联合量子研究所）；Joint Center for Quantum Information and Computer Science（量子信息与计算机科学联合中心）；National Institute of Standards and Technology（美国国家标准与技术研究院）；University of Innsbruck（因斯布鲁克大学）
An iterative algorithm for state determination is presented that uses as physical input the probability distributions for the eigenvalues of two or more observables in an unknown state $Φ$. Starting form an arbitrary state $Ψ_{0}$, a succession of states $Ψ_{n}$ is obtained that converges to $Φ$ or to a Pauli partner. This algorithm for state reconstruction is efficient and robust as is seen in the numerical tests presented and is a useful tool not only for state determination but also for the study of Pauli partners. Its main ingredient is the Physical Imposition Operator that changes any sta…
Observable-Guided Generator Selection for Improving Trainability in Quantum Machine Learning with a $ \mathfrak{g} $-Purity Interpretation under Restricted Settings
I propose a general geometric framework in which to discuss the existence of time observables. This frameworks allows one to describe a local sense in which time observables always exist, and a global sense in which they can sometimes exist subject to a restriction on the vector fields that they generate. Pauli's prohibition on quantum time observables is derived as a corollary to this result. I will then discuss how time observables can be regained in modest extensions of quantum theory beyond its standard formulation.
We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group $\langle αI, X,S\rangle$, where $α=e^{iπ/4}$ and $S=\operatorname{diag}(1,i)$. We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians etc. We also highligh…
