Stephen P. Jordan

Quantum Algorithms for Quantum Field Theories

Quantum Leap? Quantum computers are expected to be able to solve some of the most difficult problems in mathematics and physics. It is not known, however, whether quantum field theories (QFTs) can be simulated efficiently with a quantum computer. QFTs are used in particle and condensed matter physics and have an infinite number of degrees of freedom; discretization is necessary to simulate them digitally. Jordan et al. (p. 1130; see the Perspective by Hauke et al.) present an algorithm for the efficient simulation of a particular kind of QFT (with quartic interactions) and estimate the error caused by discretization. Even for the most difficult case of strong interactions, the run time of the algorithm was polynomial (rather than exponential) in parameters such as the number of particles, their energy, and the prescribed precision, making it much more efficient than the best classical algorithms. A quantum computer may be able to efficiently simulate theories used to describe particle scattering in accelerators. Quantum field theory reconciles quantum mechanics and special relativity, and plays a central role in many areas of physics. We developed a quantum algorithm to compute relativistic scattering probabilities in a massive quantum field theory with quartic self-interactions (ϕ4 theory) in spacetime of four and fewer dimensions. Its run time is polynomial in the number of particles, their energy, and the desired precision, and applies at both weak and strong coupling. In the strong-coupling and high-precision regimes, our quantum algorithm achieves exponential speedup over the fastest known classical algorithm.

Fast quantum computation at arbitrarily low energy

One version of the energy-time uncertainty principle states that the minimum time

Adiabatic optimization versus diffusion Monte Carlo methods

T_{perp}

BQP-completeness of scattering in scalar quantum field theory

for a quantum system to evolve from a given state to any orthogonal state is

Grover search and the no-signaling principle

h/(4 Delta E)

Yang–Baxter operators need quantum entanglement to distinguish knots

where

Classical Simulation of Yang-Baxter Gates

Delta E

The Fundamental Gap for a Class of Schrodinger Operators on Path and Hypercube Graphs

is the energy uncertainty. A related bound called the Margolus-Levitin theorem states that

Modulus of continuity eigenvalue bounds for homogeneous graphs and convex subgraphs with applications to quantum Hamiltonians

T_{perp} geq h/(2 E)

Black holes, quantum mechanics, and the limits of polynomial-time computability

where E is the expectation value of energy and the ground energy is taken to be zero. Many subsequent works have interpreted