Recently, Doosti et al. introduced the problem of distributed quantum state verification, where $m$ distributed nodes are given a copy of an unknown state $ρ$, and can send limited one way communication to a central node, who has a complete description of a known state $σ$. They ask how many distributed nodes $m$ are required, before the central node can succeed at distinguishing whether $ρ=σ$ or $\|ρ-σ\|_1\geq\varepsilon$ with high probability. In the setting where only quantum communication is allowed, Doosti et al. exhibit conditional lower bounds in both the public and private-coin settings, and a matching upper bound in the public-coin setting. We extend these results, and show unconditional lower bounds for when both classical and quantum communication are permitted. We show the public-coin lower bound is tight by giving an algorithm with a matching upper bound. We also show an almost tight upper bound in the private-coin setting when only quantum communication is permitted.
Comments
52 pages, 2 tables; fixed minor typos and added open problem
Recently, Doosti et al. introduced the problem of distributed quantum state verification, where m distributed nodes are given a copy of an unknown state ρ, and can send limited one way communication to a central node, who has a complete description of a known state σ. They ask how many distributed nodes m are required, before the central node can succeed at distinguishing whether ρ=σ or ∥ρ−σ∥1≥ε with high probability. In the setting where only quantum communication is allowed, Doosti et al. exhibit conditional lower bounds in both the public and private-coin settings, and a matching upper bound in the public-coin setting. We extend these results, and show unconditional lower bounds for when both classical and quantum communication are permitted. We show the public-coin lower bound is tight by giving an algorithm with a matching upper bound. We also show an almost tight upper bound in the private-coin setting when only quantum communication is permitted.
We introduce a framework for distributed quantum inference under communication constraints. In our model, m distributed nodes each receive one copy of an unknown d-dimensional quantum state ρ, before communicating via a constrained one-way communication channel with a central node, which aims to infer some property of ρ. This framework generalizes the classical distributed inference framework introduced by Acharya, Canonne, and Tyagi [COLT 2019], by allowing quantum resources such as quantum communication and shared entanglement.
Within this setting, we focus on the fundamental problem of quantum state certification: Given a complete description of some state σ, decide whether ρ=σ or ∥ρ−σ∥1≥ε. Additionally, we focus on the case of limited communication between distributed nodes and the central node: we assume each communication channel is limited to only nc bits and nq qubits with nc+nq≤logd. When all nodes can make use of a shared source of randomness, we show that the copy complexity of distributed state certification is Θ(2nq2nc/2ε2d2). We further demonstrate that shared randomness is necessary to achieve the above complexity, by proving an Ω(4nq2ncε2d3) lower bound in the private-coin setting. Moreover, we develop a private-coin algorithm that matches this bound up to a logd factor, showing this complexity is near-optimal. Together, our work establishes a general framework for distributed quantum inference with communication constraints and characterizes the complexity of distributed state certification with limited communication.
We introduce a framework for distributed quantum inference under communication constraints. In our model, m distributed nodes each receive one copy of an unknown d-dimensional quantum state ρ, before communicating via a constrained one-way communication channel with a central node, which aims to infer some property of ρ. This framework generalizes the classical distributed inference framework introduced by Acharya, Canonne, and Tyagi [COLT2019], by allowing quantum resources such as quantum communication and shared entanglement. Within this setting, we focus on the fundamental problem of quantum state certification: Given a complete description of some state σ, decide whether ρ=σ or ∥ρ−σ∥1≥ε. Additionally, we focus on the case of limited quantum communication between distributed nodes and the central node. We show that when each communication channel is limited to only nq≤logd qubits, then the sample complexity of distributed state certification is O(2nqε2d2) when public randomness is available to all nodes. Moreover, under the assumption that the channels used by the distributed nodes are mixedness-preserving, we prove a matching lower bound. We further demonstrate that shared randomness is necessary to achieve the above complexity, by proving an Ω(4nqε2d3) lower bound in the private-coin setting under the same assumption as above. Our lower bounds leverage a recently introduced quantum analogue of the celebrated Ingster-Suslina method and generalize arguments from the classical setting. Together, our work provides the first characterization of distributed quantum state certification in the regime of limited quantum communication and establishes a general framework for distributed quantum inference with communication constraints.
Quantum state certification concerns a resource-efficient way to verify whether a quantum device produces a prescribed target state without reconstructing the state in full, making it a fundamental primitive for validating quantum systems. We consider the private-coin model with only quantum communication, where the parties have no shared randomness or preshared entanglement, and no additional classical communication is allowed. We establish the optimal sample complexity, thereby closing the logarithmic gap between the previously established upper and lower bounds in this model. Remarkably, our upper bound attains the log-free optimum using a fixed family of Clifford encoders. Thus, optimal distributed state certification under private coins and quantum-only communication does not require general quantum encoders.
In this work, we consider the fundamental task of quantum state certification: given copies of an unknown quantum state ρ, test whether it matches some target state σ or is ε-far from it. For certifying d-dimensional states, Θ(d/ε2) copies of ρ are known to be necessary and sufficient. However, the algorithm achieving this complexity makes fully entangled measurements over all O(d/ε2) copies of ρ. Often, one is interested in certifying states to a high precision; this makes such joint measurements intractable even for low-dimensional states. Thus, we study whether one can obtain optimal rates for quantum state certification and related testing problems while only performing measurements on t copies at once, for some 1<t≪d/ε2. While it is well-understood how to use intermediate entanglement to achieve optimal quantum state learning, the only protocol known to achieve optimal testing is the one using fully entangled measurements.
Our main result is a smooth copy complexity upper bound for state certification as a function of t, which achieves a near-optimal rate at t=d2. In the high-precision regime, i.e., for ε<d1, this is a strict improvement over the entanglement used by the aforementioned optimal protocol. We also extend our techniques to develop new algorithms for the related tasks of mixedness testing and purity estimation, and show tradeoffs achieving the optimal rates for these problems at t=d2 as well. Our algorithms are based on novel reductions from testing to learning and leverage recent advances in quantum state tomography in a non-black-box fashion. We complement our testing upper bounds with a (nearly) matching lower bound that holds in the high-precision regime, implying that one must take t≥Ω(d2) to achieve the optimal rate for certification.
Understanding the power and limitations of classical and quantum information and how they differ is a fundamental endeavor. In property testing of distributions, a tester is given samples over a typically large domain {0,1}n. An important property is the support size both of distributions [Valiant and Valiant, STOC'11], as well, as of quantum states. Classically, even given 2n/16 samples, no tester can distinguish distributions of support size 2n/8 from 2n/4 with probability better than 2−Θ(n), even promised they are flat. Quantum states can be in a coherent superposition of states of {0,1}n, so one may ask if coherence can enhance property testing. Flat distributions naturally correspond to subset states, ∣ϕS⟩=1/∣S∣∑i∈S∣i⟩. We show that coherence alone is not enough, Coherence limitations: Given 2n/16 copies, no tester can distinguish subset states of size 2n/8 from 2n/4 with probability better than 2−Θ(n). The hardness persists even with multiple public-coin AM provers, Classical hardness with provers: Given 2O(n) samples from a distribution and 2O(n) communication with AM provers, no tester can estimate the support size up to factors 2Ω(n) with probability better than 2−Θ(n). Our result is tight. In contrast, coherent subset state proofs suffice to improve testability exponentially, Quantum advantage with certificates: With poly-many copies and subset state proofs, a tester can approximate the support size of a subset state of arbitrary size. Some structural assumption on the quantum proofs is required since we show, Collapse of QMA: A general proof cannot improve testability of any quantum property whatsoever. We also show connections to disentangler and quantum-to-quantum transformation lower bounds.
The focus of this paper is on {\em quantum distributed} computation, where we investigate whether quantum communication can help in {\em speeding up} distributed network algorithms. Our main result is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, quantum communication {\em does not} help in substantially speeding up distributed algorithms for these problems compared to the classical setting. In order to obtain this result, we extend the technique of Das Sarma et al. [SICOMP 2012] to obtain a uniform approach to prove non-trivial lower bounds for quantum distributed algorithms for several graph optimization (both exact and approximate versions) as well as verification problems, some of which are new even in the classical setting, e.g. tight randomized lower bounds for Hamiltonian cycle and spanning tree verification, answering an open problem of Das Sarma et al., and a lower bound in terms of the weight aspect ratio, matching the upper bounds of Elkin [STOC 2004]. Our approach introduces the {\em Server model} and {\em Quantum Simulation Theorem} which together provide a connection between distributed algorithms and communication complexity. The Server model is the standard two-party communication complexity model augmented with additional power; yet, most of the hardness in the two-party model is carried over to this new model. The Quantum Simulation Theorem carries this hardness further to quantum distributed computing. Our techniques, except the proof of the hardness in the Server model, require very little knowledge in quantum computing, and this can help overcoming a usual impediment in proving bounds on quantum distributed algorithms.