We consider the quantum complexity of computing Schatten p-norms and related quantities, and find that the problem of estimating these quantities is closely related to the one clean qubit model of computation. We show that the problem of approximating Tr(|A|^p) for a log-local n-qubit Hamiltonian A and p=poly(n), up to a suitable level of accuracy, is contained in DQC1; and that approximating this quantity up to a somewhat higher level of accuracy is DQC1-hard. In some cases the level of accuracy achieved by the quantum algorithm is substantially better than a natural classical algorithm for the problem. The same problem can be solved for arbitrary sparse matrices in BQP. One application of the algorithm is the approximate computation of the energy of a graph.
We consider the quantum complexity of computing Schatten p-norms and related quantities, and find that the problem of estimating these quantities is closely related to the one clean qubit model of computation. We show that the problem of approximating Tr(|A|^p) for a log-local n-qubit Hamiltonian A and p=poly(n), up to a suitable level of accuracy, is contained in DQC1; and that approximating this quantity up to a somewhat higher level of accuracy is DQC1-hard. In some cases the level of accuracy achieved by the quantum algorithm is substantially better than a natural classical algorithm for the problem. The same problem can be solved for arbitrary sparse matrices in BQP. One application of the algorithm is the approximate computation of the energy of a graph.
We study the computational complexity of estimating the normalized trace 2−nTr[f(A)] for a log-local Hamiltonian A acting on n qubits. This problem arises naturally in the DQC1 model, yet its complexity is only understood for a limited class of functions f(x).
We show that if f(x) is a continuous function with approximate degree Ω(poly(n)), then estimating 2−nTr[f(A)] up to constant additive error is DQC1-complete, under a technical condition on the polynomial approximation error of f(x). This condition holds for a broad class of functions, including exponentials, trigonometric functions, logarithms, and inverse-type functions. We further prove that when A is sparse, the classical query complexity of this problem is exponential in the approximate degree. Together, these results identify the approximate degree as the key parameter governing the complexity of normalized trace estimation: it characterizes both the quantum complexity (via efficient DQC1 algorithms) and the classical hardness, yielding an exponential quantum-classical separation. Our proof develops a unified framework that cleanly combines circuit-to-Hamiltonian constructions, periodic Jacobi operators, and tools from polynomial approximation theory, including the Chebyshev equioscillation theorem.
We consider the quantum complexity of computing Schatten p-norms and related quantities, and find that the problem of estimating these quantities is closely related to the one clean qubit model of computation. We show that the problem of approximating Tr(|A|^p) for a log-local n-qubit Hamiltonian A and p=poly(n), up to a suitable level of accuracy, is contained in DQC1; and that approximating this quantity up to a somewhat higher level of accuracy is DQC1-hard. In some cases the level of accuracy achieved by the quantum algorithm is substantially better than a natural classical algorithm for the problem. The same problem can be solved for arbitrary sparse matrices in BQP. One application of the algorithm is the approximate computation of the energy of a graph.
We investigate the dividing line between classical and quantum computational power in estimating properties of matrix functions. More precisely, we study the computational complexity of two primitive problems: given a function f and a Hermitian matrix A , we compute a matrix element of f ( A ) or a local measurement on f ( A ) | 0 ⟩ ⊗ n , with | 0 ⟩ ⊗ n being an n -qubit reference state vector, in both cases up to additive approximation error. We consider four functions—monomials, Chebyshev polynomials, the time evolution function, and the inverse function—and probe the complexity across a broad landscape covering different problem input regimes. Namely, we consider two types of matrix inputs (sparse and Pauli access), matrix properties (norm and sparsity), the approximation error, and function-specific parameters. We identify BQP- forms of both problems for each function and then toggle the problem parameters to easier regimes to see where hardness remains, or where the problem becomes classically easy. As part of our results, we make concrete a hierarchy of hardness across the functions; in parameter regimes where we have classically efficient algorithms for monomials, all three other functions remain robustly BQP-, or hard under usual computational complexity assumptions. In identifying classically easy regimes, among others, we show that for any polynomial of degree poly ( n ) , both problems can be efficiently classically simulated when A has O ( log n ) nonzero coefficients in the Pauli basis. This contrasts with the fact that the problems are BQP- in the sparse access model even for constant row sparsity, whereas the stated Pauli access efficiently constructs sparse access with row sparsity O ( log n ) . Our work provides a catalog of efficient quantum and classical algorithms for fundamental linear-algebra tasks.
Efficient measures to determine the similarity of quantum states, such as the fidelity metric, have been widely studied. In this paper, we address the problem of defining a similarity measure for quantum operations that can be efficiently estimated. Given two quantum operations, U1 and U2, represented in their circuit forms, we first develop a quantum sampling circuit to estimate the normalized Schatten 2-norm of their difference ($\ensuremath{\parallel}{U}_{1}\ensuremath{-}{U}_{2}{\ensuremath{\parallel}}_{{S}_{2}}$) with precision $\ensuremath{\epsilon}$, using only one clean qubit and one classical random variable. We prove a $\text{Poly}(\frac{1}{\ensuremath{\epsilon}})$ upper bound on the sample complexity, which is independent of the size of the quantum system. We then show that such a similarity metric is directly related to a functional definition of similarity of unitary operations using the conventional fidelity metric of quantum states (F): If $\ensuremath{\parallel}{U}_{1}\ensuremath{-}{U}_{2}{\ensuremath{\parallel}}_{{S}_{2}}$ is sufficiently small (e.g., $\ensuremath{\le}\frac{\ensuremath{\epsilon}}{1+\sqrt{2(1/\ensuremath{\delta}\ensuremath{-}1)}}$) then the fidelity of states obtained by processing the same randomly and uniformly picked pure state $|\ensuremath{\psi}\ensuremath{\rangle}$ is as high as needed $[\mathcal{F}({U}_{1}|\ensuremath{\psi}\ensuremath{\rangle},{U}_{2}|\ensuremath{\psi}\ensuremath{\rangle})\ensuremath{\ge}1\ensuremath{-}\ensuremath{\epsilon}]$ with probability exceeding $1\ensuremath{-}\ensuremath{\delta}$. We provide example applications of this efficient similarity metric estimation framework to quantum circuit learning tasks, such as finding the square root of a given unitary operation.
The local Hamiltonian (LH) problem is the canonical QMA-complete problem introduced by Kitaev. In this paper, we show its hardness in a very strong sense: we show that the 3-local Hamiltonian problem on n qubits cannot be solved classically in time O(2(1−ε)n) for any ε>0 under the Strong Exponential-Time Hypothesis (SETH), and cannot be solved quantumly in time O(2(1−ε)n/2) for any ε>0 under the Quantum Strong Exponential-Time Hypothesis (QSETH). These lower bounds give evidence that the currently known classical and quantum algorithms for LH cannot be significantly improved.
Furthermore, we are able to demonstrate fine-grained complexity lower bounds for approximating the quantum partition function (QPF) with an arbitrary constant relative error. Approximating QPF with relative error is known to be equivalent to approximately counting the dimension of the solution subspace of QMA problems. We show the SETH and QSETH hardness to estimate QPF with constant relative error. We then provide a quantum algorithm that runs in O(2n) time for an arbitrary 1/poly(n) relative error, matching our lower bounds and improving the state-of-the-art algorithm by Bravyi, Chowdhury, Gosset, and Wocjan (Nature Physics 2022) in the low-temperature regime.
To prove our fine-grained lower bounds, we introduce the first size-preserving circuit-to-Hamiltonian construction that encodes the computation of a T-time quantum circuit acting on N qubits into a (d+1)-local Hamiltonian acting on N+O(T1/d) qubits. This improves the standard construction based on the unary clock, which uses N+O(T) qubits.
Inspired by connections to two dimensional quantum theory, we define several models of computation based on permuting distinguishable particles (which we call balls), and characterize their computational complexity. In the quantum setting, we find that the computational power of this model depends on the initial input states. More precisely, with a standard basis input state, we show how to approximate the amplitudes of this model within additive error using the model DQC1 (the class of problems solvable with one clean qubit), providing evidence that the model in this case is weaker than universal quantum computing. However, for specific choices of input states, the model is shown to be universal for BQP in an encoded sense. We use representation theory of the symmetric group to partially classify the computational complexity of this model for arbitrary input states. Interestingly, we find some input states which yield a model intermediate between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an integrable scattering problem in 1+1 dimensions. We show it is universal under postselection, if we allow intermediate destructive measurements and specific input states. Therefore, the existence of any classical procedure to sample from the output distribution of this model within multiplicative error implies collapse of polynomial hierarchy to its third level. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP. Moreover, we find a nondeterministic version of this model is NP-complete.