Montanaro's polynomial representation expresses amplitudes of quantum circuits over the gates H, Z, CZ, and CCZ as normalized gaps of degree-three polynomials over 𝔽_2. The normalization is governed by the circuit width w(f), the minimum number of qubits in any circuit realizing a polynomial f. Thus, efficient width minimization would give an approximate-counting route toward a combinatorial characterization of BQP. We study the computational complexity of this parameter. For degree-three polynomials with no constant term, deciding whether w(f)≤ k is NP-complete, resolving Montanaro's open question. We also prove NP-hardness of approximation within any factor 49/48-ε, and show via a twin-copy construction that the exact and approximation hardness results also hold for degree-two polynomials. Under the Exponential Time Hypothesis, the exact problem admits no 2^o(n)-time algorithm when k=Θ(n). Complementing these hardness results, we give a nondeterministic polynomial-time search algorithm using 2log_2nk=O(klog(en/k)) witness bits, and a constructive fixed-parameter algorithm parameterized by k with running time k^6k+o(k)n+O(m).
In the search with wildcards problem [Ambainis, Montanaro, Quantum Inf. Comput.'14], one's goal is to learn an unknown bit-string x ∈{-1,1}^n. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost O(√(n)log n) and a near-matching lower bound of Ω(√(n)). Belovs [Comput. Comp.'15] subsequently showed a tight O(√(n)) upper bound. We consider a natural generalization of this problem, parametrized by a subset Q⊆ 2^[n], where an algorithm may test whether x_S = b for an arbitrary S ∈Q and b ∈{-1,1}^S of its choice, at unit cost. We show near-tight bounds when Q is any of the following collections: bounded-size sets, contiguous blocks, prefixes, and only the full set. All of these results are derived using a framework that we develop. Using symmetries of the task at hand we show that the quantum query complexity of learning x is characterized, up to a constant factor, by an optimization program, which is succinctly described as follows: `maximize over all odd functions f : {-1,1}^n →ℝ the ratio of the maximum value of f to the maximum (over T ∈Q) standard deviation of f on a subcube whose free variables are exactly T.' To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.
The Clifford hierarchy is a fundamental structure in quantum computation whose mathematical properties are not fully understood. In this work, we characterize permutation gates -- unitaries which permute the $2^n$ basis states -- in the third level of the hierarchy. We prove that any permutation gate in the third level must be a product of Toffoli gates in what we define as \emph{staircase form}, up to left and right multiplications by Clifford permutations. We then present necessary and sufficient conditions for a staircase form permutation gate to be in the third level of the Clifford hierarchy. As a corollary, we construct a family of non-semi-Clifford permutation gates $\{U_k\}_{k\geq 3}$ in staircase form such that each $U_k$ is in the third level but its inverse is not in the $k$-th level.
We present quantum speedups for sampling from distributions of the form π∝ e^-f on ℝ^d. We consider two stochastic oracle models: a stochastic gradient oracle, where f=1/n∑_i=1^n f_i and component gradients {∇ f_i}_i ∈ [n] are available, and a stochastic evaluation oracle, where only noisy values of f are available. Our framework accelerates classical stochastic Langevin Monte Carlo (LMC) and Hamiltonian Monte Carlo (HMC) algorithms by replacing stochastic gradient estimators with variance-controlled quantum mean estimation and gradient estimation subroutines. Unlike quantum walk based approaches, our algorithms do not require reversibility or exact gradients, and they preserve the structure of the underlying Markov chain. In the finite-sum setting, quantum mean estimation combined with classical variance-reduction techniques improves the stochastic gradient-query complexity for the approximate sampling task. In the stochastic zeroth-order setting, we develop gradient estimators robust to noisy function evaluations, yielding improved evaluation complexity for LMC and HMC. These results apply to strongly log-concave and/or non-log-concave distributions satisfying a log-Sobolev inequality, with convergence guarantees in Wasserstein distance and Kullback–Leibler divergence. We also show that faster sampling methods lead to quantum speedups for optimization, including for non-smooth and approximately convex objectives.
We find a modification to QMA where having one quantum proof is strictly less powerful than having two unentangled proofs, assuming EXP NEXP. This gives a new route to prove QMA(2) = NEXP that overcomes the primary drawback of a recent approach [arXiv:2402.18790 , arXiv:2306.13247] (QIP 2024). Our modification endows each proof with a form of *multipartite* unentanglement: after tracing out one register, a small number of qubits are separable from the rest of the state.
In this work we study a variant of the local Hamiltonian problem where we restrict to Hamiltonians that live on a lattice and are invariant under translations and rotations of the lattice. In the one-dimensional case this problem is known to be QMA(EXP)-complete. On the other hand, if we fix the lattice length then in the high-dimensional limit the ground state becomes unentangled due to arguments from mean-field theory. We take steps towards understanding this complexity spectrum by studying a problem that is intermediate between these two extremes. Namely, we consider the regime where the lattice dimension is arbitrary but fixed and the lattice length is scaled. We prove that this rotation-invariant Hamiltonian problem is QMA(EXP)-complete answering an open question of [6]. This characterizes a broad parameter range in which these rotation-invariant Hamiltonians have high computational complexity. 2012 ACM Subject Classification Theory of computation -> Quantum complexity theory
Space complexity is a key field of study in theoretical computer science. In the quantum setting there are clear motivations to understand the power of space-restricted computation, as qubits are an especially precious and limited resource. Recently, a new branch of space-bounded complexity called catalytic computing has shown that reusing space is a very powerful computational resource, especially for subroutines that incur little to no space overhead. While quantum catalysis in an information theoretic context, and the power of “dirty” qubits for quantum computation, has been studied over the years, these models are generally not suitable for use in quantum space-bounded algorithms, as they either rely on specific catalytic states or destroy the memory being borrowed. We define the notion of catalytic computing in the quantum setting and show a number of initial results about the model. First, we show that quantum catalytic logspace can always be computed quantumly in polynomial time; the classical analogue of this is the largest open question in catalytic computing. This also allows quantum catalytic space to be defined in an equivalent way with respect to circuits instead of Turing machines. We also prove that quantum catalytic logspace can simulate log-depth threshold circuits, a class which is known to contain (and believed to strictly contain) quantum logspace, thus showcasing the power of quantum catalytic space. Finally we show that both unitary quantum catalytic logspace and classical catalytic logspace can be simulated in the one-clean qubit model.
Previously, all known variants of the Quantum Satisfiability (QSAT) problem - consisting of determining whether a k-local (k-body) Hamiltonian is frustration-free - could be classified as being either in P; or complete for NP, MA, or QMA(1). Here, we present new qubit variants of this problem that are complete for BQP(1), coRP, QCMA, PI(coRP, NP), PI(BQP(1), NP), PI(BQP(1), MA), SoPU(coRP, NP), SoPU(BQP(1), NP), and SoPU(BQP(1), MA). Our result implies that a complete classification of quantum constraint satisfaction problems (QCSPs), analogous to Schaefer's dichotomy theorem for classical CSPs, must either include these 13 classes, or otherwise show that some are equal. Additionally, our result showcases two new types of QSAT problems that can be decided efficiently, as well as the first nontrivial BQP(1)-complete problem. We first construct QSAT problems on qudits that are complete for BQP(1), coRP, and QCMA. These are made by restricting the finite set of Hamiltonians to consist of elements similar to H-init, H-prop, and H-out, seen in the circuit-to-Hamiltonian transformation. Usually, these are used to demonstrate hardness of QSAT and Local Hamiltonian problems, and so our proofs of hardness are simple. The difficulty lies in ensuring that all Hamiltonians generated with these three elements can be decided in their respective classes. For this, we build our Hamiltonian terms with high dimensional data and clock qudits, ternary logic, and either monogamy of entanglement or specific clock encodings. We then show how to express these problems in terms of qubits, by proving that any QCSP can be reduced to a qubit problem while maintaining the same complexity - something not believed possible classically. The remaining six problems are obtained by considering "sums" and "products" of some of the QSAT problems mentioned here. Before this work, the QSAT problems generated in this way resulted in complete problems for PI and SoPU classes that were trivially equal to NP, MA, or QMA(1). We thus commence the study of these new and seemingly nontrivial classes. While [Meiburg, 2021] first sought to prove completeness for coRP, BQP(1), and QCMA, we note that those constructions are flawed. Here, we rework them, provide correct proofs, and obtain improvements on the required qudit dimensionality. 2012 ACM Subject Classification Theory of computation -> Quantum complexity theory
The (tolerant) Hamiltonian locality testing problem, introduced in [Bluhm, Caro,Oufkir `24], is to determine whether a Hamiltonian $H$ is $\varepsilon_1$-close to being $k$-local (i.e. can be written as the sum of weight-$k$ Pauli operators) or $\varepsilon_2$-far from any $k$-local Hamiltonian, given access to its time evolution operator and using as little total evolution time as possible, with distance typically defined by the normalized Frobenius norm. We give the tightest known bounds for this problem, proving an $\text{O}\left(\sqrt{\frac{\varepsilon_2}{(\varepsilon_2-\varepsilon_1)^5}}\right)$ evolution time upper bound and an $\Omega\left(\frac{1}{\varepsilon_2-\varepsilon_1}\right)$ lower bound. Our algorithm does not require reverse time evolution or controlled application of the time evolution operator, although our lower bound applies to algorithms using either tool. Furthermore, we show that if we are allowed reverse time evolution, this lower bound is tight, giving a matching $\text{O}\left(\frac{1}{\varepsilon_2-\varepsilon_1}\right)$ evolution time algorithm.
Quantum query complexity is typically characterized in terms of xor queries |x, y > bar right arrow |x, y circle plus f(x)> or phase queries, which ensure that even queries to non-invertible functions are unitary. When querying a permutation, another natural model is unitary: in-place queries |x > bar right arrow |f(x)>. Some problems are known to require exponentially fewer in-place queries than xor queries, but no separation has been shown in the opposite direction. A candidate for such a separation was the problem of inverting a permutation over N elements. This task, equivalent to unstructured search in the context of permutations, is solvable with O(root N) xor queries but was conjectured to require Omega(N) in-place queries. We refute this conjecture by designing a quantum algorithm for Permutation Inversion using O(root N) in-place queries. Our algorithm achieves the same speedup as Grover's algorithm despite the inability to efficiently uncompute queries or perform straightforward oracle-controlled reflections. Nonetheless, we show that there are indeed problems which require fewer xor queries than in-place queries. We introduce a subspace-conversion problem called Function Erasure that requires 1 xor query and Theta(root N) in-place queries. Then, we build on a recent extension of the quantum adversary method to characterize exact conditions for a decision problem to exhibit such a separation, and we propose a candidate problem. 2012 ACM Subject Classification Theory of computation -> Quantum query complexity
We study the tradeoffs between the locality and parameters of subsystem codes. We prove lower bounds on both the number and lengths of interactions in any D-dimensional embedding of a subsystem code. Specifically, we show that any embedding of a subsystem code with parameters [[ n, k, d]] into R-D must have at least M* interactions of length at least l*, where M* = Omega(max( k, d)), and l* = Omega( max (d/n(D-1/D), (kd(1/D-1)/n) (D-1/D))). We also give tradeoffs between the locality and parameters of commuting projector codes in D-dimensions, generalizing a result of Dai and Li [8]. We provide explicit constructions of embedded codes that show our bounds are optimal in both the interaction count and interaction length. 2012 ACM Subject Classification Hardware -> Quantum error correction and fault tolerance; Theory of computation -> Error-correcting codes
We show that for any fixed (2+1)-dimensional TQFT over C of either Turaev-Viro-Barrett-Westbury or Reshetikhin-Turaev type, the problem of (exactly) computing its invariants on closed 3-manifolds is either solvable in polynomial time, or else it is #P-hard to (exactly) contract certain tensors that are built from the TQFT's fusion category. Our proof is an application of a dichotomy result of Cai and Chen [J. ACM, 2017] concerning weighted constraint satisfaction problems over C. We leave for future work the issue of reinterpreting the conditions of Cai and Chen that distinguish between the two cases (i.e. #P-hard tensor contractions vs. polynomial time invariants) in terms of fusion categories. We expect that with more effort, our reduction can be improved so that one gets a dichotomy directly for TQFTs' invariants of 3-manifolds rather than more general tensors built from TQFTs' fusion categories. 2012 ACM Subject Classification Mathematics of computing -> Geometric topology; Theory of computation -> Problems, reductions and completeness; Theory of computation -> Quantum complexity theory
Quantum pseudorandomness has found applications in many areas of quantum information, ranging from entanglement theory, to models of scrambling phenomena in chaotic quantum systems, and, more recently, in the foundations of quantum cryptography. Kretschmer (TQC '21) showed that both pseudorandom states and pseudorandom unitaries exist even in a world without classical one-way functions. To this day, however, all known constructions require classical cryptographic building blocks which are themselves synonymous with the existence of one-way functions, and which are also challenging to implement on realistic quantum hardware. In this work, we seek to make progress on both of these fronts simultaneously - by decoupling quantum pseudorandomness from classical cryptography altogether. We introduce a quantum hardness assumption called the Hamiltonian Phase State (HPS) problem, which is the task of decoding output states of a random instantaneous quantum polynomial-time (IQP) circuit. Hamiltonian phase states can be generated very efficiently using only Hadamard gates, single-qubit Z rotations and CNOT circuits. We show that the hardness of our problem reduces to a worst-case version of the problem, and we provide evidence that our assumption is plausibly fully quantum; meaning, it cannot be used to construct one-way functions. We also show information-theoretic hardness when only few copies of HPS are available by proving an approximate t-design property of our ensemble. Finally, we show that our HPS assumption and its variants allow us to efficiently construct many pseudorandom quantum primitives, ranging from pseudorandom states, to quantum pseudoentanglement, to pseudorandom unitaries, and even primitives such as public-key encryption with quantum keys. 2012 ACM Subject Classification Theory of computation -> Cryptographic primitives
We study quantum algorithms for verifying properties of the output probability distribution of a classical or quantum circuit, given access to the source code that generates the distribution. We consider the basic task of uniformity testing, which is to decide if the output distribution is uniform on [d] or epsilon-far from uniform in total variation distance. More generally, we consider identity testing, which is the task of deciding if the output distribution equals a known hypothesis distribution, or is epsilon-far from it. For both problems, the previous best known upper bound was O(min{d(1/3)/epsilon(2), d(1/2)/epsilon}). Here we improve the upper bound to O(min{d(1/3)/epsilon(4/3), d(1/2)/epsilon}), which we conjecture is optimal. 2012 ACM Subject Classification Theory of computation -> Design and analysis of algorithms; Theory of computation -> Quantum computation theory
Providing evidence that quantum computers can efficiently prepare low-energy or thermal states of physically relevant interacting quantum systems is a major challenge in quantum information science. A newly developed quantum Gibbs sampling algorithm by Chen, Kastoryano, and Gilyén provides an efficient simulation of the detailed-balanced dissipative dynamics of non-commutative quantum systems. The running time of this algorithm depends on the mixing time of the corresponding quantum Markov chain, which has not been rigorously bounded except in the high-temperature regime. In this work, we establish a polylog(n) upper bound on its mixing time for various families of random n by n sparse Hamiltonians at any constant temperature. We further analyze how the choice of the jump operators for the algorithm and the spectral properties of these sparse Hamiltonians influence the mixing time. Our result places this method for Gibbs sampling on par with other efficient algorithms for preparing low-energy states of quantumly easy Hamiltonians.
We introduce a quantum cloning game in which k separate collaborative parties receive a classical input, determining which of them has to share a maximally entangled state with an additional party (referee). We provide the optimal winning probability of such a game for every number of parties k, and show that it decays exponentially when the game is played n times in parallel. These results have applications to quantum cryptography, in particular in the topic of quantum position verification, where we show security of the routing protocol (played in parallel), and a variant of it, in the random oracle model.
This work investigates the family of extended tilted-CHSH inequalities in the single-prover cryptographic compiled setting. In particular, we show that a quantum polynomial-time prover can violate these Bell inequalities by at most negligibly more than the violation achieved by two non-communicating quantum provers. To obtain this result, we extend a sum-of-squares technique to monomials with arbitrarily high degree in the Bob operators and degree at most one in the Alice operators. We also introduce a notion of partial self-testing for the compiled setting, which resembles a weaker form of self-testing in the bipartite setting. As opposed to certifying the full model, partial self-testing attempts to certify the reduced states and measurements on separate subsystems. In the compiled setting, this is akin to the states after the first round of interaction and measurements made on that state. Lastly, we show that the extended tilted-CHSH inequalities satisfy this notion of a compiled self-test. 2012 ACM Subject Classification Theory of computation -> Computational complexity and cryptography
