Recent insights concerning the PhaseLift algorithm for retrieving phases have furthered our understanding of low rank matrix recovery from rank-one projective measurements. Motivated by the structure of certain quantum mechanical experiments, we introduce a particular class of such rank-one measurements: orthonormal basis measurements. One such measurement corresponds to choosing an orthonormal basis and treating all the rank-one projectors onto different basis elements as a series of consecutive measurement matrices. We elaborate on performing low-rank matrix recovery from few, sufficiently random orthonormal basis measurements and sketch applications of such a procedure in quantum physics. We conclude this article by presenting numerical experiments testing such an approach.
Recent insights concerning the PhaseLift algorithm for retrieving phases have furthered our understanding of low rank matrix recovery from rank-one projective measurements. Motivated by the structure of certain quantum mechanical experiments, we introduce a particular class of such rank-one measurements: orthonormal basis measurements. One such measurement corresponds to choosing an orthonormal basis and treating all the rank-one projectors onto different basis elements as a series of consecutive measurement matrices. We elaborate on performing low-rank matrix recovery from few, sufficiently random orthonormal basis measurements and sketch applications of such a procedure in quantum physics. We conclude this article by presenting numerical experiments testing such an approach.
We study the recovery of low-rank Hermitian matrices from rank-one measurements obtained by uniform sampling from complex projective 3-designs, using nuclear-norm minimization. This framework includes phase retrieval as a special case via the PhaseLift method. In general, complex projective t-designs provide a practical means of partially derandomizing Gaussian measurement models. While near-optimal recovery guarantees are known for 4-designs, and it is known that 2-designs do not permit recovery with a subquadratic number of measurements, the case of 3-designs has remained open. In this work, we close this gap by establishing recovery guarantees for (exact and approximate) 3-designs that parallel the best-known results for 4-designs. In particular, we derive bounds on the number of measurements sufficient for stable and robust low-rank recovery via nuclear-norm minimization. Our results are especially relevant in practice, as explicit constructions of 4-designs are significantly more challenging than those of 3-designs.
Estimating quantum time-series such as the Loschmidt amplitude f(t)=⟨ψ∣e−iHt∣ψ⟩ is central to spectroscopy, Hamiltonian analysis, and many phase-estimation algorithms. Direct estimation via the Hadamard test requires controlled implementations of e−iHt, and the depth of these controlled circuits grows with t, making long-time estimation challenging on near-term hardware. We introduce Quantum Phaselift, a lifting-based framework that estimates the rank-one matrix Z=ff† rather than estimating f directly. We propose simple quantum circuits for estimating the entries of Z and show that measuring only a narrow band of this matrix around the diagonal is sufficient to uniquely recover f. Crucially, this reformulation decouples the controlled circuit depth from the maximum evolution time to scale instead with the width of the measured band. We prove that a O(1) bandwidth suffices for generic signals, leading to substantial savings in controlled operations compared to direct estimation methods. We develop three recovery algorithms with provable exact recovery in the noiseless setting and stability under measurement noise. Finally, we numerically demonstrate that high-quality recovery is possible for the 2D Fermi-Hubbard and 2D transverse-field Ising model signals of size exceeding 100 time points using only a few million measurement shots and reasonable post-processing time, making our time-series estimation techniques efficient and effective for near-term implementations.
We study the problem of reconstructing an unknown matrix M of rank r and dimension d using O(rd poly log d) Pauli measurements. This has applications in quantum state tomography, and is a non-commutative analogue of a well-known problem in compressed sensing: recovering a sparse vector from a few of its Fourier coefficients. We show that almost all sets of O(rd log^6 d) Pauli measurements satisfy the rank-r restricted isometry property (RIP). This implies that M can be recovered from a fixed ("universal") set of Pauli measurements, using nuclear-norm minimization (e.g., the matrix Lasso), with nearly-optimal bounds on the error. A similar result holds for any class of measurements that use an orthonormal operator basis whose elements have small operator norm. Our proof uses Dudley's inequality for Gaussian processes, together with bounds on covering numbers obtained via entropy duality.
The purpose of this paper is to introduce techniques of obtaining optimal ways to determine a [Formula: see text]-level quantum state or distinguish such states. It entails designing constrained elementary measurements extracted from maximal abelian subsets of a unitary basis [Formula: see text] for the operator algebra [Formula: see text] of a Hilbert space [Formula: see text] of finite dimension [Formula: see text] or, after choosing an orthonormal basis for [Formula: see text], for the ⋆-algebra [Formula: see text] of complex matrices of order [Formula: see text]. Illustrations are given for the techniques. It is shown that the Schwinger basis [Formula: see text] of unitary operators can give for [Formula: see text], a product of primes [Formula: see text] and [Formula: see text], the ideal number [Formula: see text] of rank one projectors that have a few quantum mechanical overlaps (or, for that matter, a few angles between the corresponding unit vectors). Finally, we give a combination of the tensor product and constrained elementary measurement techniques to deal with all [Formula: see text], though with more overlaps or angles depending on the factorization of [Formula: see text] as a product of primes or their powers like [Formula: see text] with [Formula: see text], all primes, [Formula: see text] for [Formula: see text], or other types. A comparison is drawn for different forms of unitary bases for the Hilbert space factors of the tensor product like [Formula: see text] or [Formula: see text], where [Formula: see text] is the Galois field of size [Formula: see text] and [Formula: see text] is the ring of integers modulo [Formula: see text]. Even though as Hilbert spaces they are isomorphic, but quantum mechanical system-wise, these tensor products are different. In the process, we also study the equivalence relation on unitary bases defined by R. F. Werner [J. Phys. A: Math. Gen. 34 (2001) 7081–7094], connect it to local operations on maximally entangled vectors bases, find an invariant for equivalence classes in terms of certain commuting systems, called fan representations, and, relate it to mutually unbiased bases and Hadamard matrices. Illustrations are given in the context of Latin squares and projective representations as well.
We present an algebraic algorithm for quantum state tomography that leverages measurements of certain observables to estimate structured entries of the underlying density matrix. Under low-rank assumptions, the remaining entries can be obtained solely using standard numerical linear algebra operations. The proposed algebraic matrix completion framework applies to a broad class of generic, low-rank mixed quantum states and, compared with state-of-the-art methods, is computationally efficient while providing deterministic recovery guarantees.
The process of reconstructing quantum states from experimental measurements, accomplished through quantum state tomography (QST), plays a crucial role in verifying and benchmarking quantum devices. A key challenge of QST is to find out how the accuracy of the reconstruction depends on the number of state copies used in the measurements. When multiple measurement settings are used, the total number of state copies is determined by multiplying the number of measurement settings with the number of repeated measurements for each setting. Due to statistical noise intrinsic to quantum measurements, a large number of repeated measurements is often used in practice. However, recent studies have shown that even with single-sample measurements–where only one measurement sample is obtained for each measurement setting–high accuracy QST can still be achieved with a sufficiently large number of different measurement settings. In this paper, we establish a theoretical understanding of the trade-off between the number of measurement settings and the number of repeated measurements per setting in QST. Our focus is primarily on low-rank density matrix recovery using Pauli measurements. We delve into the global landscape underlying the low-rank QST problem and demonstrate that the joint consideration of measurement settings and repeated measurements ensures a bounded recovery error for all second-order critical points, to which optimization algorithms tend to converge. This finding suggests the advantage of minimizing the number of repeated measurements per setting when the total number of state copies is held fixed. Additionally, we prove that the Wirtinger gradient descent algorithm can converge to the region of second-order critical points with a linear convergence rate. We have also performed numerical experiments to support our theoretical findings.