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 (φ(4) 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.

The computational cost of exact methods for quantum simulation using classical computers grows exponentially with system size. As a consequence, these techniques can be applied only to small systems. By contrast, we demonstrate that quantum computers could exactly simulate chemical reactions in polynomial time. Our algorithm uses the split-operator approach and explicitly simulates all electron-nuclear and interelectronic interactions in quadratic time. Surprisingly, this treatment is not only more accurate than the Born-Oppenheimer approximation but faster and more efficient as well, for all reactions with more than about four atoms. This is the case even though the entire electronic wave function is propagated on a grid with appropriately short time steps. Although the preparation and measurement of arbitrary states on a quantum computer is inefficient, here we demonstrate how to prepare states of chemical interest efficiently. We also show how to efficiently obtain chemically relevant observables, such as stateto-state transition probabilities and thermal reaction rates. Quantum computers using these techniques could outperform current classical computers with 100 qubits.electronic structure ͉ quantum computation ͉ quantum simulation A ccurate simulations of quantum-mechanical processes have greatly expanded our understanding of the fundamentals of chemical reaction dynamics. In particular, recent years have seen tremendous progress in methods development, which has enabled simulations of increasingly complex quantum systems. Although it is, strictly speaking, true that exact quantum simulation requires resources that scale exponentially with system size, several techniques are available that can treat realistic chemical problems, at a given accuracy, with only a polynomial cost. Certain fully quantum methods-such as multiconfigurational time-dependent Hartree (MCTDH) (1), matching pursuit/split-operator Fourier transform (MP/SOFT) (2), or full multiple spawning (FMS) (3)-solve the nuclear Schrödinger equation, including nonadiabatic effects, given analytic expressions for the potential energy surfaces and the couplings between them. These techniques have permitted the simulation of large systems; as examples we can give MCTDH simulations of a pentaatomic chemical reaction (4) and of a spin-boson model with 80 degrees of freedom (5) or an MP/SOFT simulation of photoisomerization in rhodopsin with 25 degrees of freedom (6). Ab initio molecular-dynamics techniques such as ab initio multiple spawning (AIMS) (7) avoid analytic expressions for potential energy surfaces and instead solve electronic Schrödinger equations at every time step. This allows one to gain insight into dynamical problems such as isomerizations through conical intersections (8).However, there are also chemical processes that are best treated by completely avoiding the Born-Oppenheimer approximation. As examples, we can cite strong-field electronic dynamics in atoms and multielectron ionization (9, 10) or atomic and molecular fragmentation cau...

All comments are subject to release under the Freedom of Information Act (FOIA).ii Reports on Computer Systems TechnologyThe Information Technology Laboratory (ITL) at the National Institute of Standards and Technology (NIST) promotes the U.S. economy and public welfare by providing technical leadership for the Nation's measurement and standards infrastructure. ITL develops tests, test methods, reference data, proof of concept implementations, and technical analyses to advance the development and productive use of information technology. ITL's responsibilities include the development of management, administrative, technical, and physical standards and guidelines for the cost-effective security and privacy of other than national security-related information in federal information systems. AbstractIn recent years, there has been a substantial amount of research on quantum computersmachines that exploit quantum mechanical phenomena to solve mathematical problems that are difficult or intractable for conventional computers. If large-scale quantum computers are ever built, they will be able to break many of the public-key cryptosystems currently in use. This would seriously compromise the confidentiality and integrity of digital communications on the Internet and elsewhere. The goal of post-quantum cryptography (also called quantum-resistant cryptography) is to develop cryptographic systems that are secure against both quantum and classical computers, and can interoperate with existing communications protocols and networks. This Internal Report shares the National Institute of Standards and Technology (NIST)'s current understanding about the status of quantum computing and post-quantum cryptography, and outlines NIST's initial plan to move forward in this space. The report also recognizes the challenge of moving to new cryptographic infrastructures and therefore emphasizes the need for agencies to focus on crypto agility.

From dice to modern electronic circuits, there have been many attempts to build better devices to generate random numbers. Randomness is fundamental to security and cryptographic systems and to safeguarding privacy. A key challenge with random-number generators is that it is hard to ensure that their outputs are unpredictable. For a random-number generator based on a physical process, such as a noisy classical system or an elementary quantum measurement, a detailed model that describes the underlying physics is necessary to assert unpredictability. Imperfections in the model compromise the integrity of the device. However, it is possible to exploit the phenomenon of quantum non-locality with a loophole-free Bell test to build a random-number generator that can produce output that is unpredictable to any adversary that is limited only by general physical principles, such as special relativity. With recent technological developments, it is now possible to carry out such a loophole-free Bell test. Here we present certified randomness obtained from a photonic Bell experiment and extract 1,024 random bits that are uniformly distributed to within 10. These random bits could not have been predicted according to any physical theory that prohibits faster-than-light (superluminal) signalling and that allows independent measurement choices. To certify and quantify the randomness, we describe a protocol that is optimized for devices that are characterized by a low per-trial violation of Bell inequalities. Future random-number generators based on loophole-free Bell tests may have a role in increasing the security and trust of our cryptographic systems and infrastructure.

Recently, there has been growing interest in using adiabatic quantum computation as an architecture for experimentally realizable quantum computers. One of the reasons for this is the idea that the energy gap should provide some inherent resistance to noise. It is now known that universal quantum computation can be achieved adiabatically using two-local Hamiltonians. The energy gap in these Hamiltonians scales as an inverse polynomial in the number of quantum gates being simulated. Here we present stabilizer codes which can be used to produce a constant energy gap against one-local and two-local noise. The corresponding fault-tolerant universal Hamiltonians are four-local and six-local, respectively, which are the optimal result achievable within this framework.

Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground-state energy of the Transverse Field Ising Model with long-range interactions with tunable range, and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot, individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground-state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward toward the application of the QAOA to more general problem instances.

The theory of causal dynamical triangulations (CDT) attempts to define a nonperturbative theory of quantum gravity as a sum over space-time geometries. One of the ingredients of the CDT framework is a global time foliation, which also plays a central role in the quantum gravity theory recently formulated by Hořava. We show that the phase diagram of CDT bears a striking resemblance with the generic Lifshitz phase diagram appealed to by Hořava. We argue that CDT might provide a unifying nonperturbative framework for anisotropic as well as isotropic theories of quantum gravity.

Quantum field theory provides the framework for the most fundamental physical theories to be confirmed experimentally and has enabled predictions of unprecedented precision. However, calculations of physical observables often require great computational complexity and can generally be performed only when the interaction strength is weak. A full understanding of the foundations and rich consequences of quantum field theory remains an outstanding challenge. We develop a quantum algorithm to compute relativistic scattering amplitudes in massive φ 4 theory in spacetime of four and fewer dimensions. The algorithm runs in a time that is polynomial in the number of particles, their energy, and the desired precision, and applies at both weak and strong coupling. Thus, it offers exponential speedup over existing classical methods at high precision or strong coupling.