In recent years simulations of chemistry and condensed materials has emerged as one of the preeminent applications of quantum computing, offering an exponential speedup for the solution of the electronic structure for certain strongly correlated electronic systems. To date, most treatments have ignored the question of whether relativistic effects, which are described most generally by quantum electrodynamics (QED), can also be simulated on a quantum computer in polynomial time. Here we show that effective QED, which is equivalent to QED to second order in perturbation theory, can be simulated in polynomial time under reasonable assumptions while properly treating all four components of the wavefunction of the fermionic field. In particular, we provide a detailed analysis of such simulations in position and momentum basis using Trotter-Suzuki formulas. We find that the number of T -gates needed to perform such simulations on a 3D lattice of ns sites scales at worst as O(ns3/ϵ)1+o(1) in the thermodynamic limit for position basis simulations and O(ns4+2/3/ϵ)1+o(1) in momentum basis. We also find that qubitization scales slightly better with a worst case scaling of O~(ns2+2/3/ϵ) for lattice eQED and complications in the prepare circuit leads to a slightly worse scaling in momentum basis of O~(ns5+2/3/ϵ) . We further provide concrete gate counts for simulating a relativistic version of the uniform electron gas that show challenging problems can be simulated using fewer than 1013 non-Clifford operations and also provide a detailed discussion of how to prepare multi-reference configuration interaction states in effective QED which can provide a reasonable initial guess for the ground state. Finally, we estimate the planewave cutoffs needed to accurately simulate heavy elements such as gold.
Abstract U(1) lattice gauge theories (LGTs) offer a way to simulate quantum electrodynamics, one of the three forces unified by the Standard Model of particles physics. Here we provide complete, quantum-gate-by-quantum-gate algorithms to simulate U(1) LGTs on a fault-tolerant quantum computer. We further perform rigorous error analysis in order to derive concrete estimates of the quantum computational resources required for an accurate simulation of U(1) LGTs using a second-order product formula. We show that U(1) LGTs in any spatial dimension can be simulated using O ˜ ( T 3 2 N 3 2 Λ ϵ − 1 2 ) non-Clifford T gates, where T is the simulation time, N is the number of lattice sites, Λ is the truncation parameter for the bosonic gauge fields, and ε is the simulation error. This work paves the way towards fault-tolerant quantum simulations of physical models closely related to the Standard Model of particle physics.
Quantum algorithms for simulating electronic ground states are slower than popular classical mean-field algorithms such as Hartree-Fock and density functional theory but offer higher accuracy. Accordingly, quantum computers have been predominantly regarded as competitors to only the most accurate and costly classical methods for treating electron correlation. However, here we tighten bounds showing that certain first-quantized quantum algorithms enable exact time evolution of electronic systems with exponentially less space and polynomially fewer operations in basis set size than conventional real-time time-dependent Hartree-Fock and density functional theory. Although the need to sample observables in the quantum algorithm reduces the speedup, we show that one can estimate all elements of the k-particle reduced density matrix with a number of samples scaling only polylogarithmically in basis set size. We also introduce a more efficient quantum algorithm for first-quantized mean-field state preparation that is likely cheaper than the cost of time evolution. We conclude that quantum speedup is most pronounced for finite-temperature simulations and suggest several practically important electron dynamics problems with potential quantum advantage.
In quantum chemistry, the "electronic structure problem" refers to the process of numerically solving Schrodinger's equation for a given molecular system. These calculations rank among the most CPU-intensive computations that are performed nowadays, yet they are also crucial to the design of new drugs and materials. The idea behind quantum simulation of quantum chemistry is that a quantum computer could perform such calculations in a manner that is exponentially more efficient than what a classical computer could accomplish. In fact, quantum simulation is quite possibly the application most naturally suited to a quantum computer. In this work we introduce novel algorithms for the simulation of quantum chemistry, algorithms that are exponentially more precise than previous algorithms in the literature. Previous algorithms are based on the Trotter-Suzuki decomposition and scale like the inverse of the desired precision, while our algorithms represent the first practical application of the recently developed truncated Taylor series approach, and they achieve scaling that is logarithmic in the inverse of the precision. We explore algorithms for both second-quantized and first-quantized encodings of the chemistry Hamiltonian, where the second-quantized encoding requires
mathcalO(N) qubits for an
N spin-orbital system, and the highly compressed first-quantized encoding requires
mathcalO(eta) qubits, where
eta<<N is the number of electrons in the system. We first introduce a second-quantized algorithm that uses pre-computed molecular integrals, resulting in a gate count that scales like
widetildecalO(N8t). Next we introduce algorithms that compute molecular integrals on the fly by carefully discretizing these integrals. Indeed, figuring out how to evaluate the integrals in an efficient manner is one of the main challenges of applying the truncated Taylor series approach to the chemistry problem: the complexity of a general quantum algorithm is traditionally formulated in terms of the number of queries made to an oracle whose assumption we just assume, but in this case, we actually construct the oracle for our specific problem. The evaluation of the oracle itself requires evaluating molecular integrals, which adds additional complexity to an algorithm. We find that the gate count of the second-quantized on-the-fly algorithm scales like
widetildecalO(N5t). We also find that the first-quantized on-the-fly algorithm, which uses the configuration interaction (CI) matrix representation of the Hamiltonian, requires at most
widetildecalO(eta2N3t) gates. In general, the discovery of exponentially more precise algorithms will allow for the simulation of larger molecular systems than was previously possible under Trotter-based methods. In addition, the results here can be generalized to the simulation of other fermionic systems beyond the electronic structure problem.
The Schwinger model (quantum electrodynamics in 1+1 dimensions) is a testbed for the study of quantum gauge field theories. We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In particular, we perform a tight analysis of low-order Trotter formula simulations of the Schwinger model, using recently derived commutator bounds, and give upper bounds on the resources needed for simulations in both scenarios. In lattice units, we find a Schwinger model on N / 2 physical sites with coupling constant x − 1 / 2 and electric field cutoff x − 1 / 2 Λ can be simulated on a quantum computer for time 2 x T using a number of T -gates or CNOTs in O ~ ( N 3 / 2 T 3 / 2 x Λ ) for fixed operator error. This scaling with the truncation Λ is better than that expected from algorithms such as qubitization or QDRIFT. Furthermore, we give scalable measurement schemes and algorithms to estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable–the mean pair density. Finally, we bound the root-mean-square error in estimating this observable via simulation as a function of the diamond distance between the ideal and actual CNOT channels. This work provides a rigorous analysis of simulating the Schwinger model, while also providing benchmarks against which subsequent simulation algorithms can be tested.
We describe methods to construct digital quantum simulation algorithms for quantum spin systems on a regular lattice with local interactions. In addition to tools such as the Trotter-Suzuki expansion and graph coloring, we also discuss the efficiency gained by parallel execution of an extensive number of commuting terms. We provide resource estimates and quantum circuit elements for the most important cases and classes of spin systems. As resource estimates we indicate the total number of gates N N and simulation time T T , expressed in terms of the number n n of spin 1/2 lattice sites (qubits), target accuracy \epsilon ϵ , and simulated time t t . We provide circuit constructions that realize the simulation time T^{(1)}\propto nt^2/\epsilon T ( 1 ) ∝ n t 2 / ϵ and T^{(2q)}\propto t^{1+\eta}n^\eta/\epsilon^\eta T ( 2 q ) ∝ t 1 + η n η / ϵ η for arbitrarily small \eta = 1/2q η = 1 / 2 q for the first-order and higher-order Trotter expansions. We also discuss the potential impact of scaled gates, which have not yet been fully explored.
We provide an explicit recursive divide-and-conquer approach for simulating quantum dynamics and derive a discrete first-quantized nonrelativistic QED Hamiltonian based on the many-particle Pauli-Fierz Hamiltonian. We apply this recursive divide-and-conquer algorithm to this Hamiltonian and compare it to a concrete simulation algorithm that uses qubitization. Our divide-and-conquer algorithm, using lowest-order Trotterization, scales for fixed grid spacing as O ~ ( Λ N 2 η 2 t 2 / ϵ ) for grid size N , η particles, simulation time t , field cutoff Λ , and error ϵ . Our qubitization algorithm scales as O ~ ( N ( η + N ) ( η + Λ 2 ) t log ( 1 / ϵ ) ) . This shows that even a naive partitioning and low-order splitting formula can yield, through our divide-and-conquer formalism, superior scaling to qubitization for large Λ . We compare the relative costs of these two algorithms on systems that are relevant for applications such as the spontaneous emission of photons and the photoionization of electrons. We observe that for different parameter regimes, one method can be favored over the other. Finally, we give new algorithmic and circuit-level techniques for gate optimization, including a new way of implementing a group of multicontrolled- X gates that can be used for better analysis of circuit cost. Published by the American Physical Society 2024