Two Combinatorial MA-Complete Problems

Despite the interest in the complexity class MA, the randomized analog of NP, just a few natural MA-complete problems are known. The first problem was found by (Bravyi and Terhal, SIAM Journal of Computing 2009); it was then followed by (Crosson, Bacon and Brown, PRE 2010) and (Bravyi, Quantum Information and Computation 2015). Surprisingly, two of these problems are defined using terminology from quantum computation, while the third is inspired by quantum computation and keeps a physical terminology. This prevents classical complexity theorists from studying these problems, delaying potential progress, e.g., on the NP vs. MA question. Here, we define two new combinatorial problems and prove their MA-completeness. The first problem, ACAC, gets as input a succinctly described graph, with some marked vertices. The problem is to decide whether there is a connected component with only unmarked vertices, or the graph is far from having this property. The second problem, SetCSP, generalizes standard constraint satisfaction problem (CSP) into constraints involving sets of strings. Technically, our proof that SetCSP is MA-complete is based on an observation by (Aharonov and Grilo, FOCS 2019), in which it was noted that a restricted case of Bravyi and Terhal's problem (namely, the uniform case) is already MA-complete; a simple trick allows to state this restricted case using combinatorial language. The fact that the first, more natural, problem of ACAC is MA-hard follows quite naturally from this proof, while the containment of ACAC in MA is based on the theory of random walks. We notice that the main result of Aharonov and Grilo carries over to the SetCSP problem in a straightforward way, implying that finding a gap-amplification procedure for SetCSP (as in Dinur's PCP proof) is equivalent to MA=NP. This provides an alternative new path towards the major problem of derandomizing MA.Comment: Minor changes; Published in the proceedings of ITCS 202

Guidable Local Hamiltonian Problems with Implications to Heuristic Ans\"atze State Preparation and the Quantum PCP Conjecture

We study 'Merlinized' versions of the recently defined Guided Local Hamiltonian problem, which we call 'Guidable Local Hamiltonian' problems. Unlike their guided counterparts, these problems do not have a guiding state provided as a part of the input, but merely come with the promise that one exists. We consider in particular two classes of guiding states: those that can be prepared efficiently by a quantum circuit; and those belonging to a class of quantum states we call classically evaluatable, for which it is possible to efficiently compute expectation values of local observables classically. We show that guidable local Hamiltonian problems for both classes of guiding states are $\mathsf{QCMA}$-complete in the inverse-polynomial precision setting, but lie within $\mathsf{NP}$ (or $\mathsf{NqP}$) in the constant precision regime when the guiding state is classically evaluatable. Our completeness results show that, from a complexity-theoretic perspective, classical Ans\"atze selected by classical heuristics are just as powerful as quantum Ans\"atze prepared by quantum heuristics, as long as one has access to quantum phase estimation. In relation to the quantum PCP conjecture, we (i) define a complexity class capturing quantum-classical probabilistically checkable proof systems and show that it is contained in $\mathsf{BQP}^{\mathsf{NP}[1]}$ for constant proof queries; (ii) give a no-go result on 'dequantizing' the known quantum reduction which maps a $\mathsf{QPCP}$-verification circuit to a local Hamiltonian with constant promise gap; (iii) give several no-go results for the existence of quantum gap amplification procedures that preserve certain ground state properties; and (iv) propose two conjectures that can be viewed as stronger versions of the NLTS theorem. Finally, we show that many of our results can be directly modified to obtain similar results for the class $\mathsf{MA}$.Comment: 61 pages, 6 figure

Spectral estimation for stoquastic Hamiltonians: a comparison between classical imaginary-time evolution and quantum real-time evolution

We present a classical Monte Carlo (MC) scheme which efficiently estimates an imaginary-time, decaying signal for stoquastic (i.e. sign-problem-free) local Hamiltonians. The decay rates in this signal correspond to Hamiltonian eigenvalues (with associated eigenstates present in an initial state) and can be classically extracted using a classical signal processing method like ESPRIT. We compare the efficiency of this MC scheme to its quantum counterpart in which one extracts eigenvalues of a general local Hamiltonian from a real-time, oscillatory signal obtained through quantum phase estimation circuits, again using the ESPRIT method. We prove that the ESPRIT method can resolve S = poly(n) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) quantum and classical effort through the quantum phase estimation circuits. We prove that our Monte Carlo scheme plus the ESPRIT method can resolve S = O(1) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) purely classical effort for stoquastic Hamiltonians. These results thus quantify some opportunities and limitations of classical Monte Carlo methods for spectral estimation of stoquastic Hamiltonians. We numerically compare these MC and QPE eigenvalue estimation schemes by implementing them for an archetypal stoquastic Hamiltonian system: the transverse field Ising chain

A classical oracle separation between QMA and QCMA

It is a long-standing open question in quantum complexity theory whether the definition of $\textit{non-deterministic}$ quantum computation requires quantum witnesses $(\textsf{QMA})$ or if classical witnesses suffice $(\textsf{QCMA})$. We make progress on this question by constructing a randomized classical oracle separating the respective computational complexity classes. Previous separations [Aaronson-Kuperberg (CCC'07), Fefferman-Kimmel (MFCS'18)] required a quantum unitary oracle. The separating problem is deciding whether a distribution supported on regular un-directed graphs either consists of multiple connected components (yes instances) or consists of one expanding connected component (no instances) where the graph is given in an adjacency-list format by the oracle. Therefore, the oracle is a distribution over $n$-bit boolean functions.Comment: 43 page

Complexity of the Guided Local Hamiltonian Problem: Improved Parameters and Extension to Excited States

Recently it was shown that the so-called guided local Hamiltonian problem -- estimating the smallest eigenvalue of a $k$-local Hamiltonian when provided with a description of a quantum state ('guiding state') that is guaranteed to have substantial overlap with the true groundstate -- is BQP-complete for $k \geq 6$ when the required precision is inverse polynomial in the system size $n$, and remains hard even when the overlap of the guiding state with the groundstate is close to a constant $\left(\frac12 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)\right)$. We improve upon this result in three ways: by showing that it remains BQP-complete when i) the Hamiltonian is 2-local, ii) the overlap between the guiding state and target eigenstate is as large as $1 - \Omega\left(\frac{1}{\mathop{poly}(n)}\right)$, and iii) when one is interested in estimating energies of excited states, rather than just the groundstate. Interestingly, iii) is only made possible by first showing that ii) holds.Comment: 22 pages, 3 figure

Spectral estimation for Hamiltonians: A comparison between classical imaginary-time evolution and quantum real-time evolution

We consider the task of spectral estimation of local quantum Hamiltonians. The spectral estimation is performed by estimating the oscillation frequencies or decay rates of signals representing the time evolution of states. We present a classical Monte Carlo (MC) scheme which efficiently estimates an imaginary-time, decaying signal for stoquastic (i.e. sign-problem-free) local Hamiltonians. The decay rates in this signal correspond to Hamiltonian eigenvalues (with associated eigenstates present in an input state) and can be extracted using a classical signal processing method like ESPRIT. We compare the efficiency of this MC scheme to its quantum counterpart in which one extracts eigenvalues of a general local Hamiltonian from a real-time, oscillatory signal obtained through quantum phase estimation circuits, again using the ESPRIT method. We prove that the ESPRIT method can resolve S = poly(n) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) quantum and classical effort through the quantum phase estimation circuits, assuming efficient preparation of the input state. We prove that our Monte Carlo scheme plus the ESPRIT method can resolve S = O(1) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) purely classical effort for stoquastic Hamiltonians, requiring some access structure to the input state. However, we also show that under these assumptions, i.e. S = O(1) eigenvalues, assuming a 1/poly(n) gap between them and some access structure to the input state, one can achieve this with poly(n) purely classical effort for general local Hamiltonians. These results thus quantify some opportunities and limitations of Monte Carlo methods for spectral estimation of Hamiltonians. We numerically compare the MC eigenvalue estimation scheme (for stoquastic Hamiltonians) and the quantum-phase-estimation-based eigenvalue estimation scheme by implementing them for an archetypal stoquastic Hamiltonian system: the transverse field Ising chain

Assessing, testing, and challenging the computational power of quantum devices

Randomness is an intrinsic feature of quantum theory. The outcome of any measurement will be random, sampled from a probability distribution that is defined by the measured quantum state. The task of sampling from a prescribed probability distribution therefore seems to be a natural technological application of quantum devices. And indeed, certain random sampling tasks have been proposed to experimentally demonstrate the speedup of quantum over classical computation, so-called “quantum computational supremacy”. In the research presented in this thesis, I investigate the complexity-theoretic and physical foundations of quantum sampling algorithms. Using the theory of computational complexity, I assess the computational power of natural quantum simulators and close loopholes in the complexity-theoretic argument for the classical intractability of quantum samplers (Part I). In particular, I prove anticoncentration for quantum circuit families that give rise to a 2-design and review methods for proving average-case hardness. I present quantum random sampling schemes that are tailored to large-scale quantum simulation hardware but at the same time rise up to the highest standard in terms of their complexity-theoretic underpinning. Using methods from property testing and quantum system identification, I shed light on the question, how and under which conditions quantum sampling devices can be tested or verified in regimes that are not simulable on classical computers (Part II). I present a no-go result that prevents efficient verification of quantum random sampling schemes as well as approaches using which this no-go result can be circumvented. In particular, I develop fully efficient verification protocols in what I call the measurement-device-dependent scenario in which single-qubit measurements are assumed to function with high accuracy. Finally, I try to understand the physical mechanisms governing the computational boundary between classical and quantum computing devices by challenging their computational power using tools from computational physics and the theory of computational complexity (Part III). I develop efficiently computable measures of the infamous Monte Carlo sign problem and assess those measures both in terms of their practicability as a tool for alleviating or easing the sign problem and the computational complexity of this task. An overarching theme of the thesis is the quantum sign problem which arises due to destructive interference between paths – an intrinsically quantum effect. The (non-)existence of a sign problem takes on the role as a criterion which delineates the boundary between classical and quantum computing devices. I begin the thesis by identifying the quantum sign problem as a root of the computational intractability of quantum output probabilities. It turns out that the intricate structure of the probability distributions the sign problem gives rise to, prohibits their verification from few samples. In an ironic twist, I show that assessing the intrinsic sign problem of a quantum system is again an intractable problem