This paper studies the quantum identity testing problem, a quantum analogue of distribution identity testing. The goal is to determine whether a quantum state is identical to another fixed quantum state, using as few state samples as possible. Rather than general entangled measurements, we consider the less powerful but experimentally friendly Pauli measurements. We follow the standard setting where Pauli measurements are regarded as two-outcome measurements, i.e., each$n$-qubit Pauli measurement has a one-bit outcome. We prove that for an$n$-qubit quantum system, the sample complexity of this problem is$\Theta \left({\mathrm {poly}(n)\cdot \frac {4^{n}}{\epsilon ^{2}}}\right)$if only Pauli measurements are allowed. In other words, we provide simple algorithms to determine whether two$n$-qubit quantum states,$\rho $and$\sigma $, are identical or$\epsilon $-far in trace distance using Pauli measurements, using$O\left({\min \{n^{4},n^{3}+\log ^{3}(1/\epsilon)\}\cdot \frac {4^{n}}{\epsilon ^{2}}}\right)$copies of$\rho $and$\sigma $. Interestingly,$\mathcal {O}\left({\frac {4^{n}}{\epsilon ^{2}}}\right)$copies are not sufficient under this setting.
This paper studies the quantum identity testing problem, a quantum analogue of distribution identity testing. The goal is to determine whether a quantum state is identical to another fixed quantum state, using as few state samples as possible. Rather than general entangled measurements, we consider the less powerful but experimentally friendly Pauli measurements. We follow the standard setting where Pauli measurements are regarded as two-outcome measurements, i.e., eachn-qubit Pauli measurement has a one-bit outcome. We prove that for ann-qubit quantum system, the sample complexity of this problem isΘ(poly(n)⋅ϵ24n)if only Pauli measurements are allowed. In other words, we provide simple algorithms to determine whether twon-qubit quantum states,ρandσ, are identical orϵ-far in trace distance using Pauli measurements, usingO(min{n4,n3+log3(1/ϵ)}⋅ϵ24n)copies ofρandσ. Interestingly,O(ϵ24n)copies are not sufficient under this setting.
In this paper, we show that Θ(poly(n)⋅ε24n) is the sample complexity of testing whether two n-qubit quantum states ρ and σ are identical or ε-far in trace distance using two-outcome Pauli measurements.
We study the following quantum state identity problem: given access to copies of n unknown pure quantum states, the goal is to determine whether every pair has fidelity at least 1−ε or some pair has fidelity at most ε. This problem relaxes the identical-or-orthogonal promise considered in previous work. For any fixed overlap parameter ε∈(0,1/4), we present two quantum algorithms with complementary guarantees on circuit depth and sample complexity. The first runs in constant depth using parallel swap tests and takes O(log(n/δ)log(2/δ)) copies of each input state to succeed with probability at least 1−δ. The second uses the Schur transform to reduce the sample complexity to O(log(n/δ)), through an analysis relating Jucys--Murphy elements to Schur sampling. We also prove a matching sample complexity lower bound of Ω(log(n/δ)) for any quantum algorithm with failure probability at most δ≤1/2−c, for any constant c>0, establishing that the second algorithm achieves the optimal dependence on both n and δ for fixed ε.
Pauli Measurements are the most important measurements in both theoretical and experimental aspects of quantum information science. In this paper, we explore the power of Pauli measurements in the state tomography related problems. Firstly, we show that the \textit{quantum state tomography} problem of n-qubit system can be accomplished with O(ε210n) copies of the unknown state using Pauli measurements. As a direct application, we studied the \textit{quantum overlapping tomography} problem introduced by Cotler and Wilczek in Ref. \cite{Cotler_2020}. We show that the sample complexity is O(ε210k⋅log((kn)/δ))) for quantum overlapping tomography of k-qubit reduced density matrices among n is quantum system, where 1−δ is the confidential level, and ε is the trace distance error. This can be achieved using Pauli measurements. Moreover, we prove that Ω(ε2log(n/δ)) copies are needed. In other words, for constant k, joint, highly entangled, measurements are not asymptotically more efficient than Pauli measurements.
We consider a fundamental problem of \emph{mixedness testing}: Given n copies of an N-qubit state ρ, determine whether ρ=Id/d or ∥ρ−Id/d∥1≥ε with high probability, where d=2N. In particular, we focus on performing this task in the practical setting of single-qubit measurements, where measurements are prepared independently on each qubit. We provide a nearly complete picture of single-qubit mixedness tesing by showing n=Θ~(10N/ε2). To establish our lower bound, we introduce a measurement-dependent lower bound framework for adaptive single-copy state certification. For the upper bound, we present a randomized Pauli basis measurement protocol, which relies on a new primitive for computationally efficient uniformity testing of correlation-concentrated distributions on the Boolean hypercube. In conjunction, we provide lower and upper bounds for mixedness testing with fixed Pauli measurement protocols.
We revisit the problem of Pauli shadow tomography: given copies of an unknown n-qubit quantum stateρ, estimate Tr(Pρ)for some set of Pauli operatorsFto within additive errorϵ. This has been a popular testbed for exploring the advantage of protocols with quantum memory over those without: with enough memory to measure two copies at a time, one can use Bell sampling to estimate∣Tr(Pρ)for allPusingO(n/ϵ4)copies, but withk≤nqubits of memory,Ω(2(n−k)/3)copies are needed. These results leave open several natural questions. How does this picture change in the physically relevant setting where one only needs to estimate a certain subset of Paulis? What is the optimal dependence onϵ?What is the optimal tradeoff between quantum memory and sample complexity? We answer all of these questions: •For any subsetAof Paulis and any family of measurement strategies, we completely characterize the optimal sample complexity, up tolog∣A∣factors. •We show any protocol that makes poly(n)-copy measure-ments must makeΩ(1/ϵ4)measurements. •For any protocol that makes poly(n)-copy measurements and only hask<nqubits of memory, we show thatΘ~(min{2n/ϵ2,2n−k/ϵ4})copies are necessary and sufficient. The protocols we propose can also estimate the actual valuesTr(Pρ), rather than just their absolute values as in prior work. Additionally, as a byproduct of our techniques, we establish tight bounds for the task of purity testing and show that it exhibits an intriguing phase transition not present in the memory-sample tradeoff for Pauli shadow tomography.
We study the problems of quantum state certification, equivalence testing and independence testing. In certification, given samples of an unknown quantum state ρ and the description of a state σ, the goal is to test whether ρ=σ, or whether ρ and σ are far in a given distance measure. In equivalence testing, σ is also unknown and only accessible via samples. Independence testing decides whether ρAC=ρA⊗ρC, or is far from being a product. The sample complexities of these problems are now well-understood for a decision gap ε in trace distance: in the single-copy measurement setting with d-dimensional states, all three tasks can be solved using the same non-adaptive approach, which uses Θ(d3/2/ε2) samples and is optimal in general, even without adaptivity.
In this work, we consider decision gaps expressed in fidelity and study possible separations between these problems and how adaptivity can help. We prove that certification with respect to fidelity for a state σ of rank r does not benefit from adaptivity and requires Θ(r3/2/ε) samples. For equivalence testing and independence testing, we provide adaptive algorithms using O(min{d3/2/ε2,d9/4/ε}) and O(min{(dAdC)3/2/ε2,dA9/4dC3/4/ε}) samples, for dA≥dC, respectively. Our main technique is a framework that uses partial learning and a reduction to testing in ℓ2-distance, adapted from the distribution testing literature. We show that adaptivity matters for equivalence testing in fidelity by proving that Ω(1/ε2) samples are necessary in the non-adaptive case even for qubits, showing a separation from certification.