Algorithms 4

Abstract: We present a quantum algorithm for simulating the time evolution generated by any bounded, time-dependent operator $-A$ with non-positive logarithmic norm, thereby serving as a natural generalization of the Hamiltonian simulation problem. Our method generalizes the recent Linear-Combination-of-Hamiltonian-Simulation (LCHS) framework. In instances where $A$ is time-independent, we provide a block-encoding of the evolution operator $e^{-At}$ with $\mathcal{O}\big(t\log\frac{1}{\epsilon})$ queries to the block-encoding oracle for $A$. We also show how the normalized evolved state can be prepared with $\mathcal{O}(1/\|e^{-At}\ket{\vec{u}_0}\|)$ queries to the oracle that prepares the normalized initial state $\ket{\vec{u}_0}$. These complexities are optimal in all parameters and improve the error scaling over prior results. Furthermore, we show that any improvement of our approach exceeding a constant factor of approximately 3 is infeasible. For general time-dependent operators $A$, we also prove that a uniform trapezoidal rule on our LCHS construction yields exponential convergence, leading to simplified quantum circuits with improved gate complexity compared to prior nonuniform-quadrature methods.

Abstract: The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We present a suite of methods to do matrix arithmetics---with the result encoded in the off-diagonal blocks of a Hamiltonian---using Hamiltonian evolutions of input operators. We show how to maintain this Hamiltonian block encoding, so that matrix operations can be composed one after another, and the entire computation takes $\leq 2$ ancilla qubits. We achieve this for matrix multiplication, matrix addition, matrix inversion, Hermitian conjugation, fractional scaling, integer scaling, complex phase scaling, as well as singular value transformation for both odd and even polynomials. We also develop an overlap estimation algorithm to extract classical properties of Hamiltonian block encoded operators, analogous to the well known Hadmard test, at no extra cost of qubit. Our Hamiltonian matrix multiplication uses the Lie group commutator product formula and its higher-order generalizations due to Childs and Wiebe. Our Hamiltonian singular value transformation employs a dominated polynomial approximation, wherein the approximation holds over the domain of interest, while the polynomial is upper bounded by the target function on the entire unit interval. When applied to quantum simulation, our methods inherit the commutator scaling of product formulas, while leveraging the power of matrix arithmetics to reduce the cost of each step. To illustrate this feature, we present a circuit for simulating a class of sum-of-squares Hamiltonians, covering systems studied recently in quantum chemistry. Our simulation attains a commutator scaling in step count, while the gate cost per step remains comparable to that of more advanced algorithms. We achieve this with $1$ ancilla qubit.

Abstract: We present sum-of-squares spectral amplification (SOSSA), a framework for improving quantum simulation relevant to low-energy problems. We show how SOSSA can be applied to problems like energy and phase estimation and provide fast quantum algorithms for these problems that significantly improve over prior art. We analyze the performance of SOSSA on the Sachdev-Ye-Kitaev model, a representative strongly correlated system, and demonstrate asymptotic speedups over generic simulation methods by a factor of the square root of the system size. We then apply SOSSA to electronic structure problems in quantum chemistry, yielding a factor of 4 to 195 speedup over the state of the art in ground-state energy estimation for models of Iron-Sulfur complexes and a CO2-fixation catalyst.

Abstract: We show a relation between quantum learning theory and algorithmic hardness. We demonstrate that finding near-ground states of certain sparse disordered quantum systems is average-case hard for Lipschitz quantum algorithms if there exists an efficient, local learning algorithm---such as the classical shadows algorithm---for estimating the energy of a state of the system. A corollary of our result is that many standard quantum algorithms fail to find near-ground states of these systems, including short-time Lindbladian dynamics, short-time quantum annealing, phase estimation, and shallow-depth variational quantum algorithms. Our results are unconditional. To achieve this, we introduce a topological property of quantum systems that we call the quantum overlap gap property (QOGP). This property is only satisfied by systems with an efficient local learning algorithm for the energy. We prove that systems which exhibit this topological property in their low-energy space are intractable for quantum algorithms whose outputs are stable under perturbations to their inputs. We then prove that the QOGP is satisfied for a sparsified variant of the quantum p-spin model, giving the first known algorithmic hardness-of-approximation result for quantum algorithms in finding the ground state of a non-stoquastic, noncommuting quantum system. Our resulting lower bound for quantum algorithms optimizing this model using Lindbladian evolution matches (up to constant factors) the best-known time lower bound for classical Langevin dynamics optimizing classical p-spin models. For this reason we suspect that finding ground states of typical quantum p-spin models using quantum algorithms is, in practice, as intractable as the classical p-spin model is for classical algorithms. Inversely, we show that the Sachdev--Ye--Kitaev (SYK) model does not exhibit the QOGP, consistent with previous evidence that the model is rapidly mixing at low temperatures.