Simulations of Many-Body Quantum Systems by a Quantum Computer

Stephen Wiesner

REVIEW

Simulations of Many-Body Quantum Systems by a Quantum Computer

Stephen Wiesner

Quantum Physics · Submitted 26 Mar 1996 · 8 references

Pith's verdict: The origin paper for digital quantum simulation: the construction is right and the polynomial claim is the right claim, but it is asserted rather than derived, and the step operator as written is not unitary.

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arXiv IDquant-ph/9603028 (version 1)

Pith IDD7NB6EKF, Pith's permanent record of this paper and its review

CitationStephen Wiesner. "Simulations of Many-Body Quantum Systems by a Quantum Computer." arXiv:quant-ph/9603028 (1996). https://pith.science/paper/D7NB6EKF

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Editor's note

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This is the paper people should be citing when they say Feynman sketched the idea and someone else made it an algorithm. Wiesner gives two explicit constructions: Trotter-stepping a sum of M one- and two-body terms on N qubits, and a position/momentum encoding with a Fourier transform on each particle's register to handle P²/2m. Both are the right constructions, and both are still what people actually implement. The second one in particular — k qubits per particle, FFT to momentum, advance the phase, FFT back — is the template everyone has been filling in for thirty years. That is real credit and it should be given plainly. The paper is also honest about its scope: one dimension, no symmetrization, bounded box, stated up front rather than buried. I don't count those as weaknesses; they're declared simplifications in a three-page note. The reader's report is right on the substance. The load-bearing claim is polynomial time, and the only convergence argument in the text is 'Δt can be chosen small enough so that further decreasing of Δt does not produce a significant effect.' No bound ties Δt, T, the operator norms, and a target error together, so the step count is not shown to stay polynomial as T grows. The stress-test's concrete catch is also correct and worth naming: 'we act on the qubits with the M unitary operators I + iHΔt/ħ' — that operator is not unitary, its norm exceeds one, and it does not preserve the state norm. It's a first-order-in-Δt slip, trivially repaired with the Hermitian conjugate or a symmetrized product, and it doesn't threaten the conclusion. But it is a genuine error in the text, not a stylistic quibble. The potential-phase computation and the measurement sampling cost are named in a sentence each and never costed; again ordinary incompleteness for a first proposal, not a hidden flaw. Nothing here is circular, nothing is fitted, and the citations are prior-art anchors doing exactly the work they're cited for. Who should read it: anyone working on Hamiltonian simulation who wants the actual origin, and anyone writing a history or motivation section. It is not a resource estimate and shouldn't be cited as one. Send it to referees — the gaps are the kind a referee report would sharpen rather than reject over.

Referee report

Significance

If the constructions hold, the paper establishes that exact simulation of quantum many-body dynamics is a natural and physically motivated application of a quantum computer, with cost polynomial in particle number where the exact classical cost is exponential. The two architectures proposed are the right ones and are recognisably the ancestors of the standard approaches: Trotterised products of local Hamiltonians on a qubit register, and the first-quantised grid representation with a transform between position and momentum bases for the kinetic operator. The observation that register entanglement can stand in for target entanglement is the correct core insight, and the paper states it cleanly. The note is, however, a sketch: it contains no derivation of the polynomial bound it advertises, no gate counts, no error analysis, and no proofs, so its contribution is architectural rather than quantitative. Credit is due for identifying both simulation strategies and for the correct high-level complexity comparison; the assessment below concerns what is needed to make the advertised claim stand.

Major objections (4)

[First example, p. 1] The step operator is stated as 'the M unitary operators I + iHΔt/ħ'. This operator is not unitary: for any nonzero H its norm exceeds 1, so it does not preserve the norm of the state and is not a physical gate. The intended object is presumably the exponential e^{-iHΔt/ħ}, or a symmetrised product such as (I + iHΔt/ħ)(I - iHΔt/ħ) followed by renormalisation, which is unitary up to O(Δt²). As written, the first example's elementary step is not a valid quantum operation, and the polynomial-time claim built on it is not established by the text. This is a local, repairable error, but it is a concrete error in the load-bearing construction and should be corrected explicitly rather than left to the reader.
[First and second examples, pp. 1-2] The only convergence argument in the paper is the sentence 'Δt can be chosen small enough so that further decreasing of Δt does not produce a significant effect on the result.' No bound relates Δt, the total time T, the operator norms of the Hamiltonians, and the target error ε, and no statement is made about how the number of steps scales when T grows at fixed ε. For a first-order (Lie) splitting the error per step is O(Δt²) and the accumulated error over T/Δt steps is O(TΔt), so at fixed ε the step count grows with T; for a symmetrised (Strang) splitting the accumulated error is O(TΔt²) and the step count grows as T^{3/2}. The text's phrasing, 'we of course repeat the whole process T/Δt times', suggests a fixed Δt independent of T, which would let the error grow without bound as T increases. The polynomial-time claim is the paper's central assertion and it is asserted, not derived; a short derivation giving the gate count as a function of N, T, the Hamiltonian norms, and ε is needed. The same remark applies to the ordering of the M factors: the Hamiltonians in general do not commute, and the dependence of the splitting error on their commutators is not mentioned.
[Second example, p. 2] Two gaps in the second construction need to be closed. First, the fast Fourier transform on the k position qubits implicitly imposes periodic boundary conditions on each particle's coordinate, so the simulated system is a ring of circumference L, not a particle confined to a region of length L as stated; the two differ in their spectra and in the treatment of the potential near the edges, and the text should say which problem is being solved. Second, the potential step is described as computing V_ij from the position registers and advancing the phase by ΔtV_ij/ħ, but this requires additional ancilla workspace, reversible arithmetic to evaluate V_ij to a specified number of bits, a controlled phase rotation, and uncomputation of the ancillas; none of this appears in the cost accounting, and for a Coulomb potential the register width needed to resolve 1/r at small separation is a real constraint on the claimed polynomial in k. The claim that 'the whole calculation can be done in time polynomial in N and k' should be supported by an explicit accounting of these steps.
[Abstract and p. 1] The comparison that grounds the word 'impracticable' is against exact state-vector simulation, which needs 2^N (or 2^{Nk}) amplitudes. That is the worst case, and the text should say so: for many systems of physical interest, classical methods that exploit structure (mean-field and coupled-cluster approaches, quantum Monte Carlo where the sign problem is absent, density-matrix renormalisation for low-entanglement states) do far better than brute-force state-vector propagation, and the regimes in which they fail are precisely the strongly correlated, highly entangled ones. Scoping the claim to exact simulation of generic entangled states would make the comparison accurate and would also sharpen what the quantum computer is claimed to add.

Minor objections (6)

[Second example, p. 2] The paper explicitly states that it will not antisymmetrise or symmetrise the wavefunction. This is acknowledged, and for the motivating case of electrons in an atom it is a substantial restriction, since the unantisymmetrised simulation solves a distinguishable-particle problem rather than the fermionic one. It would help the reader to say what changes if antisymmetrisation is imposed, or at least to note that the polynomial bound is claimed only for the distinguishable-particle case.
[First example, p. 1] The number M of Hamiltonians is never related to N. For one- and two-body Hamiltonians M scales as N², which is polynomial and harmless, but stating it would make the complexity claim self-contained.
[First example, p. 1] The statement that averages are obtained 'by repeating the whole process a number of times' does not say how many repetitions are needed for a given statistical accuracy, nor how the initial state is prepared. Both are standard but should be mentioned for the resource claim to be complete.
[References] Reference [5] cites Feynman's Optics News piece; the more directly relevant citation for simulating physics with computers is Feynman, International Journal of Theoretical Physics 21, 467 (1982). Reference [6] misspells Deutsch as 'Deutch', and reference [7] misspells Alamitos as 'Alomitos'.
[Second example, p. 2] The text says the transform 'can be done in k log k time[7, 8]'; the cited sources give the transform for factoring, and the gate count for the transform on k qubits in this context should be stated in the paper's own terms rather than left to the citation.
[Second example, p. 2] The sentence 'The whole calculation can be done it time polynomial in N and k' contains a typographical error ('it' for 'in').

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The paper

The paper claims

Wiesner argues a quantum computer can simulate quantum many-body systems in polynomial time, because its entangled qubits can mimic the entangled states of the system itself.

The reading

The paper's claim is that a quantum computer can do what a classical computer cannot: follow the time evolution of a quantum many-body system without ever writing down its exponentially large state vector. Wiesner gives two constructions. In the first, $N$ two-state systems under $M$ one- and two-body Hamiltonians are represented by $N$ qubits, and the evolution is built from repeated short-time factors $I + iHΔt/ħ$; a classical machine doing the same bookkeeping would have to track a vector in $2^N$ dimensions. In the second, $N$ particles moving in one dimension have each wave function stored in $k$ qubits, with the amplitude for a given position encoded as a binary number in those qubits; kinetic energy is applied by Fourier-transforming to momentum, advancing the phase by $Δt(P^2/2mħ)$, and transforming back, while position-dependent potentials advance the phase by $ΔtV_{ij}/ħ$ evaluated on the qubit registers. The reason this matters is stated plainly: the entangled states of the quantum computer can mimic the entangled states of the quantum many-body system, so the cost grows polynomially in $N$ and $k$ instead of exponentially. If right, systems that resist first-principles classical treatment, such as complex atoms and molecules, become calculable.

What carries the argument

Two pieces of machinery carry the argument. The first is the correspondence between the computer's register and the physical state: $N$ qubits for $N$ two-state systems, or $k$ qubits per particle for a discretised wave function, with basis states labelled by binary position. The second is the replacement of a differential operator by a sequence of unitary phase advances: the split-step application of $I + iHΔt/ħ$ for each term of the Hamiltonian, and, in the particle case, the Fourier transform between position and momentum bases so that the kinetic term becomes a diagonal phase $Δt(P^2/2mħ)$ in momentum space. The load-bearing point is that the phase advance is applied to a superposition, so one hardware operation acts on all $2^{Nk}$ amplitudes at once; that is where the exponential saving comes from.

Extended reading notes

Core claim

The paper proposes that quantum simulation is a natural application of a quantum computer, and sketches two concrete schemes for it. The first is a qubit register of $N$ two-state systems acted on by $M$ one- and two-body Hamiltonians, advanced by a small time step $Δt$ through the operators $I + iHΔt/ħ$ and repeated $T/Δt$ times, with measurements repeated to accumulate averages. The second is a particle register: each of $N$ particles moving in a region of length $L$ gets $k$ qubits, with $Δx = L/2^k$ and the amplitude for position $73L/2^k$ equal to the amplitude that those qubits hold the binary number 73. The $P^2/2m$ part of the motion is applied by a fast Fourier transform on each particle's qubits (costing $k log k$), a phase advance by $Δt(P^2/2mħ)$, and a transform back; the two-particle potentials $V_{ij}$ are applied by reading the positions off the qubits of particles $i$ and $j$ and advancing the phase by $ΔtV_{ij}/ħ$, which, because the registers are in superposition, advances the whole state by a superposition of phases. The author's conclusion is that the whole calculation runs in time polynomial in $N$ and $k$, whereas the straightforward classical integration of the same equations would require tracking $2^{Nk}$ amplitudes.

Load-bearing premise

The load-bearing premise is that repeating the short-time factors $I + iHΔt/ħ$ for $T/Δt$ rounds approximates the true evolution to a fixed accuracy while the total number of operations stays polynomial in $N$ and $k$; the paper asserts that $Δt$ can be made small enough that further shrinking changes nothing, but supplies no bound linking $Δt$, $T$, the sizes of the Hamiltonian terms, and the final error.

Editorial extensions

If this is right

Complex atoms and molecules, whose electronic structure cannot be computed accurately from first principles classically, become in-principle calculable on a machine whose size is linear in the particle number.
The classical cost of the same calculation scales as $2^N$ for the qubit example and $2^{Nk}$ for the particle example, so the gap between the two approaches widens exponentially with problem size.
The scheme is general over one- and two-body Hamiltonians, so it is not tied to any particular physical system or interaction.
Because the simulation ends in measurement and repetition, expectation values of observables at time $T$ are recoverable by running the whole procedure many times.
The Fourier-transform step makes the kinetic energy cheap in the quantum register, at $k log k$ per particle, which is what keeps the position-space scheme polynomial rather than exponential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The step operator as written, $I + iHΔt/ħ$, is not unitary on its own; the natural repair is to pair it with its Hermitian conjugate or use a symmetric second-order splitting, and whether the polynomial claim survives depends on how the error of that splitting accumulates over $T/Δt$ steps.
The paper prices only the forward evolution; recovering averages requires repeating the run, so the true cost carries a prefactor set by the desired statistical precision, and that prefactor is not estimated here.
Read forward, this is the template that later digital quantum-simulation algorithms formalise: encode the state in qubits, split the Hamiltonian into simulable pieces, alternate short phase advances, and bound the Trotter error. The open item left on the table is precisely that bound.
A direct test of the particle scheme would be to simulate two or three interacting particles in one dimension at small $k$ and compare against exact diagonalisation, checking whether the error at a fixed gate budget falls as $Δt$ or $Δt^2$.

Original abstract

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

The compiled record

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.14 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

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

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We suggest that quantum computers can solve quantum many-body problems that are impracticable to solve on a classical computer.	contribution 0.74	0.15	0.80	hedged possibility 0.93

What kind of paper the classifier reports this is: position or commentary 0.92 · evidence from argument 0.87.

What was examined, and what was not

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arXiv listing abstract v1

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S sections
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (8)

[1]

R. Landauer, IBM J. Res. Dev. 5, 183 (1961)

View on Pith 1961
[2]

C.H. Bennett, IBM J. Res. Dev. 17 ,525 (1973)

View on Pith 1973
[3]

P. Benioﬀ, J. Stat. Phys. 29, 515 (1982)

View on Pith 1982
[4]

A. Peres, Phys. Rev. A 32, 3266 (1985)

View on Pith 1985
[5]

R.P. Feynman, Optics News 11, 11 (1985)

View on Pith 1985
[6]

D. Deutch and R. Jozsa, Proc. R. Soc. Lond. A 439, 553 (1992)

View on Pith 1992
[7]

P.W. Shor, Proc. of the 35th Annual Symposium on the Found ations of Computer Science (IEEE Computer Society Press, Los Alomitos CA, 1994 ), p.124

View on Pith 1994
[8]

D. Coppersmith, “An approximate Fourier transform usef ul in quantum factoring”, IBM Research Report RC19642 (1994). 3

View on Pith 1994

Forward citations

Cited by 17 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Quantum Computation of Scattering in Scalar Quantum Field Theories
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A quantum algorithm computes scattering amplitudes in massive phi-fourth theory in four or fewer spacetime dimensions in time polynomial in particle number, energy, and precision, at both weak and strong coupling.
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A quantum algorithm computes relativistic scattering probabilities in phi-fourth theory in four or fewer spacetime dimensions in polynomial time, at both weak and strong coupling.
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quant-ph 2025-05 conditional novelty 7.0 of 10

Short-time dynamics of geometrically local classical systems are dequantized and shown BPP-complete, while long-time dynamics are shown as powerful as exponential-time quantum computation.
High-precision quantum algorithms for partial differential equations
quant-ph 2020-02 conditional novelty 7.0 of 10

Quantum algorithms based on adaptive-order finite differences and spectral methods solve linear PDEs with complexity polynomial in the spatial dimension and logarithmic in one over the error tolerance, improving the e...
Low Depth Quantum Simulation of Electronic Structure
quant-ph 2017-05 accept novelty 7.0 of 10

A dual plane wave basis reduces the electronic structure Hamiltonian to O(N^2) terms and enables quantum simulation circuits with O(N^{7/2}) depth, a large asymptotic improvement over prior Gaussian-basis methods.

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A variational hybrid continuous-variable discrete-variable quantum algorithm for adiabatic nuclear dynamics
physics.chem-ph 2026-08 conditional novelty 6.0 of 10

A variational hybrid qubit-oscillator algorithm evolves frozen Gaussian wavepackets for nuclear dynamics, converging to exact results on harmonic and Morse potentials and showing partial success on double-well bifurcation.
Hardware-efficient variational quantum algorithms for time evolution
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The paper introduces cone, block, and angle update rules for variational time evolution that avoid matrix inversion and reduce hardware requirements via causal cones.
Digital quantum computation of fermion-boson interacting systems
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A finite-grid qubit encoding of harmonic-oscillator low-energy states, justified by the Nyquist-Shannon sampling theorem, enables polynomial-cost digital quantum simulation of fermion-boson systems with roughly six qu...
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quant-ph 2018-02 conditional novelty 6.0 of 10

A finite-grid qubit encoding of phonon low-energy states, justified by the Nyquist-Shannon theorem, lets quantum computers simulate electron-phonon systems with a handful of extra qubits per mode and modest circuit depth.
Quantum Algorithms for Quantum Chemistry based on the sparsity of the CI-matrix
quant-ph 2013-12 conditional novelty 5.0 of 10

Encoding the electronic Hamiltonian in a fixed-electron-number determinant basis and simulating it as a sparse matrix reduces wavefunction qubits to the logarithm of the determinant count and alters the step-count sca...
Medium induced jet broadening in a quantum computer
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A 10-11 qubit quantum circuit reproduces the known analytical jet quenching parameter for a single quark in a stochastic color medium, establishing a baseline for future quantum simulations of in-medium jet physics.
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A speculative proposal to replace many-body Hilbert-space quantum mechanics with classical swarm algorithms, including a heuristic derivation of Born's rule from an ad hoc amplitude cutoff.
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A NERSC analysis finds that more than 50% of its workload could ultimately benefit from quantum computing and that vendor roadmaps and quantum application requirements are projected to overlap in the next 5 to 10 years.
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The review surveys quantum algorithms for ground-state, dynamics, and thermal-state simulation of molecules and materials, and argues that quantum advantage is problem-specific rather than generic.
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A comprehensive survey of quantum algorithms for quantum chemistry, organized by problem type (static versus dynamic) and hardware regime (fault-tolerant versus noisy near-term).
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A comprehensive review that maps classical electronic structure methods onto quantum algorithms, with worked examples for H2 and LiH and resource estimates for near-term and fault-tolerant regimes.

How the literature uses it

54 arXiv papers cite this in 77 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 41 sentences without a confident label

Uses its method or tool 6 papers

The simulation method described here was introduced in [ref] .

Quantum Computation of Scattering in Scalar Quantum Field Theories · Simulating Hamiltonian Time Evolution · 0.91
They provide a detailed method for exact simulation of atomic and molecular electronic wavefunctions, based on discretizing the position in space, and evolving the wavefunction using the QFT-based time evolution technique presented by Wiesner [ref] and Zalka [ref] described in section [ref].

Using Quantum Computers for Quantum Simulation · Quantum chemistry · 0.96
The computational primitive used in this framework is instead the hamiltonian simulation in continuous space representation [ref] .

Sampling, rates, and reaction currents through reverse stochastic quantization on quantum computers · Introduction · 0.83

Cites it as background 27 papers

First quantized quantum simulations date back to [ref] .

Quantum simulation of exact electron dynamics can be more efficient than classical mean-field methods · Exact quantum dynamics in first quantization · 0.94
This conjecture has spurred the construction of a number of quantum algorithms to efficiently simulate quantum systems under a Hamiltonian [ref] .

Simulating Quantum Dynamics On A Quantum Computer · Introduction · 0.95
Since then, efficient quantum algorithms have been developed to simulate the dynamics of quantum lattice models and quantum systems with a fixed number of particles [ref] , but the question about quantum field theories has remained open.

Quantum Algorithms for Quantum Field Theories · Introduction · 0.97

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

R. Landauer, IBM J. Res. Dev. 5, 183 (1961)

View on Pith 1961

[2] [2]

C.H. Bennett, IBM J. Res. Dev. 17 ,525 (1973)

View on Pith 1973

[3] [3]

P. Benioﬀ, J. Stat. Phys. 29, 515 (1982)

View on Pith 1982

[4] [4]

A. Peres, Phys. Rev. A 32, 3266 (1985)

View on Pith 1985

[5] [5]

R.P. Feynman, Optics News 11, 11 (1985)

View on Pith 1985

[6] [6]

D. Deutch and R. Jozsa, Proc. R. Soc. Lond. A 439, 553 (1992)

View on Pith 1992

[7] [7]

P.W. Shor, Proc. of the 35th Annual Symposium on the Found ations of Computer Science (IEEE Computer Society Press, Los Alomitos CA, 1994 ), p.124

View on Pith 1994

[8] [8]

D. Coppersmith, “An approximate Fourier transform usef ul in quantum factoring”, IBM Research Report RC19642 (1994). 3

View on Pith 1994