Rolando Somma

We present a new primitive for quantum algorithms that implements a discrete Hermite transform efficiently, in time that depends logarithmically in both the dimension and the inverse of the allowable error. This transform, which maps basis states to states whose amplitudes are proportional to the Hermite functions, can be interpreted as the Gaussian analogue of the Fourier transform. Our algorithm is based on a method to exponentially fast forward the evolution of the quantum harmonic oscillator, which significantly improves over prior art. We apply this Hermite transform to give examples of provable quantum query advantage in property testing and learning. In particular, we show how to efficiently test the property of being close to a low-degree in the Hermite basis when inputs are sampled from the Gaussian distribution, and how to solve a Gaussian analogue of the Goldreich-Levin learning task efficiently. We also comment on other potential uses of this transform to simulating time dynamics of quantum systems in the continuum.

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.

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 We study the problem of simulating the dynamics of spin systems when the initial state is supported on a subspace of low energy of a Hamiltonian $H$. This is a central problem in physics with vast applications in many-body systems and beyond, where the interesting physics takes place in the low-energy sector. We analyze error bounds induced by product formulas that approximate the evolution operator and show that these bounds depend on an effective low-energy norm of $H$. We find improvements over the best previous complexities of product formulas that apply to the general case, and these improvements are more significant for long evolution times that scale with the system size and/or small approximation errors. To obtain these improvements, we prove novel exponentially-decaying upper bounds on the leakage to high-energy subspaces due to the product formula. Our results provide a path to a systematic study of Hamiltonian simulation at low energies, which will be required to push quantum simulation closer to reality.

We present algorithms to evaluate partition functions of quantum Hamiltonians using mixed-state quantum computation, specifically the DQC1 model of computation which requires only one pure qubit. Our algorithm provides an additive-error estimate to the normalized partition function in time that is polynomial in the number of qubits, for a large class of Hamiltonians. This implies that a variant of the partition function problem, that was previously shown to be hard for the complexity class DQC1 by Brandao, is in fact DQC1-complete. Our algorithm is based on approximations of the exponential operator as linear combinations of unitaries, which are related to block-encoding of Hamiltonians or Hamiltonian evolutions. We then extend our results to give an algorithm that estimates the partition function within a desired relative error. Towards this end, we develop a procedure based on a sequence of approximations within predetermined additive errors that may be of independent interest.