Direct certification of a class of quantum simulations

One of the main challenges in the field of quantum simulation and computation is to identify ways to certify the correct functioning of a device when a classical efficient simulation is not available. Important cases are situations in which one cannot classically calculate local expectation values of state preparations efficiently. In this work, we develop weak-membership formulations of the certification of ground state preparations. We provide a non-interactive protocol for certifying ground states of frustration-free Hamiltonians based on simple energy measurements of local Hamiltonian terms. This certification protocol can be applied to classically intractable analog quantum simulations: For example, using Feynman-Kitaev Hamiltonians, one can encode universal quantum computation in such ground states. Moreover, our certification protocol is applicable to ground states encodings of IQP circuits demonstration of quantum supremacy. These can be certified efficiently when the error is polynomially bounded.Comment: 10 pages, corrected a small error in Eqs. (2) and (5

Bounding the joint numerical range of Pauli strings by graph parameters

The interplay between the quantum state space and a specific set of measurements can be effectively captured by examining the set of jointly attainable expectation values. This set is commonly referred to as the (convex) joint numerical range. In this work, we explore geometric properties of this construct for measurements represented by tensor products of Pauli observables, also known as Pauli strings. The structure of pairwise commutation and anticommutation relations among a set of Pauli strings determines a graph $G$, sometimes also called the frustration graph. We investigate the connection between the parameters of this graph and the structure of minimal ellipsoids encompassing the joint numerical range. Such an outer approximation can be very practical since ellipsoids can be handled analytically even in high dimensions. We find counterexamples to a conjecture from [C. de Gois, K. Hansenne and O. G\"uhne, arXiv:2207.02197], and answer an open question in [M. B. Hastings and R. O'Donnell, Proc. STOC 2022, pp. 776-789], which implies a new graph parameter that we call $\beta(G)$. Besides, we develop this approach in different directions, such as comparison with graph-theoretic approaches in other fields, applications in quantum information theory, numerical methods, properties of the new graph parameter, etc. Our approach suggests many open questions that we discuss briefly at the end.Comment: 14+3 pages, 5 figure

Tripartite-to-Bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. We study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication (SLOCC). It is known that the tripartite-to-bipartite SLOCC convertibility is characterized by the maximal Schmidt rank of the given tripartite state, i.e. the largest Schmidt rank over those bipartite states lying in the support of the reduced density operator. In this paper, we further study this problem and exhibit novel results in both multi-copy and asymptotic settings, utilizing powerful results from the structure of matrix spaces. In the multi-copy regime, we observe that the maximal Schmidt rank is strictly super-multiplicative, i.e. the maximal Schmidt rank of the tensor product of two tripartite pure states can be strictly larger than the product of their maximal Schmidt ranks. We then provide a full characterization of those tripartite states whose maximal Schmidt rank is strictly super-multiplicative when taking tensor product with itself. Notice that such tripartite states admit strict advantages in tripartite-to-bipartite SLOCC transformation when multiple copies are provided. In the asymptotic setting, we focus on determining the tripartite-to-bipartite SLOCC entanglement transformation rate. Computing this rate turns out to be equivalent to computing the asymptotic maximal Schmidt rank of the tripartite state, defined as the regularization of its maximal Schmidt rank. Despite the difficulty caused by the super-multiplicative property, we provide explicit formulas for evaluating the asymptotic maximal Schmidt ranks of two important families of tripartite pure states by resorting to certain results of the structure of matrix spaces, including the study of matrix semi-invariants. These formulas turn out to be powerful enough to give a sufficient and necessary condition to determine whether a given tripartite pure state can be transformed to the bipartite maximally entangled state under SLOCC, in the asymptotic setting. Applying the recent progress on the non-commutative rank problem, we can verify this condition in deterministic polynomial time

Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while usually only use a constant number of ancilla qubits. We show that singular value transformation leads to novel algorithms. We give an efficient solution to a certain "non-commutative" measurement problem and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression. "Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, including: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms. In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.Comment: 67 pages, 1 figur

Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics

An n-qubit quantum circuit performs a unitary operation on an exponentially large, 2n-dimensional, Hilbert space, which is a major source of quantum speed-ups. We develop a new “Quantum singular value transformation” algorithm that can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator. The transformations are realized by quantum circuits with a very simple structure – typically using only a constant number of ancilla qubits – leading to optimal algorithms with appealing constant factors. We show that our framework allows describing many quantum algorithms on a high level, and enables remarkably concise proofs for many prominent quantum algorithms, ranging from optimal Hamiltonian simulation to various quantum machine learning applications. We also devise a new singular vector transformation algorithm, describe how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum, and show how to efficiently implement principal component regression. Finally, we also prove a quantum lower bound on spectral transformations

Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential ad

Polynomial Time Algorithms in Invariant Theory for Torus Actions

An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years. In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory