Quantum simulation is a prominent application of quantum computers. While there is extensive previous work on simulating finite-dimensional systems, less is known about quantum algorithms for real-space dynamics. We conduct a systematic study of such algorithms. In particular, we show that the dynamics of a d -dimensional Schrödinger equation with η particles can be simulated with gate complexity O ~ ( η d F poly ( log ( g ′ / ϵ ) ) ) , where ϵ is the discretization error, g ′ controls the higher-order derivatives of the wave function, and F measures the time-integrated strength of the potential. Compared to the best previous results, this exponentially improves the dependence on ϵ and g ′ from poly ( g ′ / ϵ ) to poly ( log ( g ′ / ϵ ) ) and polynomially improves the dependence on T and d , while maintaining best known performance with respect to η . For the case of Coulomb interactions, we give an algorithm using η 3 ( d + η ) T poly ( log ( η d T g ′ / ( Δ ϵ ) ) ) / Δ one- and two-qubit gates, and another using η 3 ( 4 d ) d / 2 T poly ( log ( η d T g ′ / ( Δ ϵ ) ) ) / Δ one- and two-qubit gates and QRAM operations, where T is the evolution time and the parameter Δ regulates the unbounded Coulomb interaction. We give applications to several computational problems, including faster real-space simulation of quantum chemistry, rigorous analysis of discretization error for simulation of a uniform electron gas, and a quadratic improvement to a quantum algorithm for escaping saddle points in nonconvex optimization.
more »
« less
Quantum walks and Dirac cellular automata on a programmable trapped-ion quantum computer
The quantum walk formalism is a widely used and highly successful framework for modeling quantum systems, such as simulations of the Dirac equation, different dynamics in both the low and high energy regime, and for developing a wide range of quantum algorithms. Here we present the circuit-based implementation of a discrete-time quantum walk in position space on a five-qubit trapped-ion quantum processor. We encode the space of walker positions in particular multi-qubit states and program the system to operate with different quantum walk parameters, experimentally realizing a Dirac cellular automaton with tunable mass parameter. The quantum walk circuits and position state mapping scale favorably to a larger model and physical systems, allowing the implementation of any algorithm based on discrete-time quantum walks algorithm and the dynamics associated with the discretized version of the Dirac equation.
more » « less- NSF-PAR ID:
- 10175049
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Nature Communications
- Volume:
- 11
- Issue:
- 1
- ISSN:
- 2041-1723
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
more » « less
Abstract Single-qubit gates are essential components of a universal quantum computer. Without selective addressing of individual qubits, scalable implementation of quantum algorithms is extremely challenging. When the qubits are discrete points or regions on a lattice, selectively addressing magnetic spin qubits at the nanoscale remains a challenge due to the difficulty of localizing and confining a classical divergence-free field to a small volume of space. Herein we propose a technique for addressing spin qubits using voltage-control of nanoscale magnetism, exemplified by the use of voltage control of magnetic anisotropy. We show that by tuning the frequency of the nanomagnet’s electric field drive to the Larmor frequency of the spins confined to a nanoscale volume, and by modulating the phase of the drive, single-qubit quantum gates with fidelities approaching those for fault-tolerant quantum computing can be implemented. Such single-qubit gate operations require only tens of femto-Joules per gate operation and have lossless, purely magnetic field control. Their physical realization is also straightforward using foundry manufacturing techniques.
-
The time-marching strategy, which propagates the solution from one time step to the next, is a natural strategy for solving time-dependent differential equations on classical computers, as well as for solving the Hamiltonian simulation problem on quantum computers. For more general homogeneous linear differential equations d d t | ψ ( t ) ⟩ = A ( t ) | ψ ( t ) ⟩ , | ψ ( 0 ) ⟩ = | ψ 0 ⟩ , a time-marching based quantum solver can suffer from exponentially vanishing success probability with respect to the number of time steps and is thus considered impractical. We solve this problem by repeatedly invoking a technique called the uniform singular value amplification, and the overall success probability can be lower bounded by a quantity that is independent of the number of time steps. The success probability can be further improved using a compression gadget lemma. This provides a path of designing quantum differential equation solvers that is alternative to those based on quantum linear systems algorithms (QLSA). We demonstrate the performance of the time-marching strategy with a high-order integrator based on the truncated Dyson series. The complexity of the algorithm depends linearly on the amplification ratio, which quantifies the deviation from a unitary dynamics. We prove that the linear dependence on the amplification ratio attains the query complexity lower bound and thus cannot be improved in the worst case. This algorithm also surpasses existing QLSA based solvers in three aspects: (1) A ( t ) does not need to be diagonalizable. (2) A ( t ) can be non-smooth, and is only of bounded variation. (3) It can use fewer queries to the initial state | ψ 0 ⟩ . Finally, we demonstrate the time-marching strategy with a first-order truncated Magnus series, which simplifies the implementation compared to high-order truncated Dyson series approach, while retaining the aforementioned benefits. Our analysis also raises some open questions concerning the differences between time-marching and QLSA based methods for solving differential equations.more » « less
-
Quantum networks are complex systems formed by the interaction among quantum processors through quantum channels. Analogous to classical computer networks, quantum networks allow for the distribution of quantum computation among quantum computers. In this work, we describe a quantum walk protocol to perform distributed quantum computing in a quantum network. The protocol uses a quantum walk as a quantum control signal to perform distributed quantum operations. We consider a generalization of the discrete-time coined quantum walk model that accounts for the interaction between a quantum walker system in the network graph with quantum registers inside the network nodes. The protocol logically captures distributed quantum computing, abstracting hardware implementation and the transmission of quantum information through channels. Control signal transmission is mapped to the propagation of the walker system across the network, while interactions between the control layer and the quantum registers are embedded into the application of coin operators. We demonstrate how to use the quantum walker system to perform a distributed CNOT operation, which shows the universality of the protocol for distributed quantum computing. Furthermore, we apply the protocol to the task of entanglement distribution in a quantum network.more » « less
-
more » « less
Abstract Numerical techniques to efficiently model out-of-equilibrium dynamics in interacting quantum many-body systems are key for advancing our capability to harness and understand complex quantum matter. Here we propose a new numerical approach which we refer to as generalized discrete truncated Wigner approximation (GDTWA). It is based on a discrete semi-classical phase space sampling and allows to investigate quantum dynamics in lattice spin systems with arbitrary
S ≥ 1/2. We show that the GDTWA can accurately simulate dynamics of large ensembles in arbitrary dimensions. We apply it forS > 1/2 spin-models with dipolar long-range interactions, a scenario arising in recent experiments with magnetic atoms. We show that the method can capture beyond mean-field effects, not only at short times, but it also can correctly reproduce long time quantum-thermalization dynamics. We benchmark the method with exact diagonalization in small systems, with perturbation theory for short times, and with analytical predictions made for models which feature quantum-thermalization at long times. We apply our method to study dynamics in largeS > 1/2 spin-models and compute experimentally accessible observables such as Zeeman level populations, contrast of spin coherence, spin squeezing, and entanglement quantified by single-spin Renyi entropies. We reveal that largeS systems can feature larger entanglement than correspondingS = 1/2 systems. Our analyses demonstrate that the GDTWA can be a powerful tool for modeling complex spin dynamics in regimes where other state-of-the art numerical methods fail.
