[Paper Review] Simulations of Many-Body Quantum Systems by a Quantum Computer

Stephen Wiesner|ArXiv.org|Mar 26, 1996

Quantum Computing Algorithms and Architecture128 citations

TL;DR

This paper proposes that quantum computers can efficiently simulate quantum many-body systems—such as atoms and molecules—by leveraging qubit entanglement to represent complex quantum states. Unlike classical computers, which face exponential resource scaling, the quantum approach achieves polynomial-time simulation by applying unitary evolution steps via Hamiltonian decomposition, enabling accurate, scalable quantum dynamics simulation.

ABSTRACT

We suggest that quantum computers can solve quantum many-body problems that are impracticable to solve on a classical computer.

Motivation & Objective

• To demonstrate that quantum computers can efficiently simulate quantum many-body systems that are intractable for classical computers.
• To address the fundamental limitation of classical simulation, which scales exponentially with system size due to the need to track $2^N$-dimensional Hilbert spaces.
• To propose a practical quantum algorithm for simulating systems governed by one- and two-body Hamiltonians using qubit-based state representation and time-evolution operators.
• To show that the simulation time scales polynomially in system size, contrasting with classical exponential scaling.

Proposed method

• Represent each quantum particle in the system using $k$ qubits to discretize its spatial wave function into $2^k$ position bins.
• Apply time-evolution operators $I + iH\Delta t/\hbar$ for each Hamiltonian term (one- and two-body) in sequence, using small time steps $\Delta t$.
• Use fast Fourier transforms (FFTs) on qubit registers to switch between position and momentum representations, enabling efficient $P^2/2m$ term evolution.
• Apply position-dependent potential terms $V_{ij}$ by computing their values from the current qubit states and applying phase shifts proportional to $\Delta t V_{ij}/\hbar$.
• Repeat the full time-evolution sequence $T/\Delta t$ times to simulate dynamics over time $T$, then measure the final state to extract observables.
• Leverage quantum superposition and entanglement to naturally represent and evolve many-body wave functions without explicitly storing the full amplitude vector.

Experimental results

Research questions

• RQ1 Can quantum computers simulate quantum many-body systems more efficiently than classical computers?
• RQ2 What is the computational complexity of simulating time evolution for a system of $N$ particles with one- and two-body interactions?
• RQ3 How can the Schrödinger equation for interacting particles be efficiently implemented on a quantum computer using qubit registers?
• RQ4 Can the exponential Hilbert space of many-body systems be simulated in polynomial time using quantum algorithms?
• RQ5 What role does entanglement play in enabling efficient simulation of quantum dynamics on a quantum computer?

Key findings

• The simulation of $N$ two-state systems under $M$ one- and two-body Hamiltonians can be performed in time polynomial in $N$, avoiding the classical exponential scaling.
• For $N$ particles in one dimension with $k$ qubits per particle, the total simulation time scales polynomially in $N$ and $k$, specifically $O(Nk \log k)$ per time step.
• The use of FFTs enables efficient momentum-space evolution, reducing the cost of kinetic energy terms from $O(2^k)$ to $O(k \log k)$ per particle.
• The method naturally handles superpositions of position states and applies phase shifts based on entangled configurations, preserving quantum coherence.
• Measurement outcomes are obtained by repeating the simulation and averaging results, yielding accurate expectation values for observables.
• The approach avoids the need to store the full $2^{Nk}$-dimensional amplitude vector, which is infeasible for classical computers.