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Adiabatic Quantum State Generation and Statistical Zero Knowledge

Dorit Aharonov, Amnon Ta-Shma

arXiv:quant-ph/0301023v2quant-ph

TL;DR

The paper approaches quantum algorithm design through quantum state generation, using adiabatic evolutions and Hamiltonian tools. It proves equivalence with circuit-model state generation and derives applications to statistical zero knowledge and Markov-chain sampling.

  • Problem

    Developing qualitatively different techniques for quantum algorithms remains difficult, motivating a state-generation perspective.

    Source: §1 · p. 2

  • Method

    The paper defines adiabatic state generation over general paths of simulatable Hamiltonians and develops projection and spectral-gap tools for constructing such evolutions.

    Source: §1 · pp. 5–6

  • Results

    Adiabatic state generation is equivalent to efficient circuit-model state generation, while statistical zero-knowledge problems reduce to CQS and slowly varying Markov chains yield quantum sampling algorithms.

    Source: §1 · pp. 3, 6, 7

  • Takeaways & Limitations

    The framework links quantum state generation with statistical zero knowledge and approximate-counting procedures based on slowly varying Markov chains.

    Source: §1 · pp. 3, 7

    Using one delay parameter T may appear restrictive, although the paper states that more complicated schedules can be encoded through the dependence on s.

    Source: §4 · p. 18

Abstract

from arXiv · show

The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to the problem, by studying the problem of 'quantum state generation'. This approach provides intriguing links between many different areas: quantum computation, adiabatic evolution, analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing Markov chains, the complexity class statistical zero knowledge, quantum random walks, and more. We first show that many natural candidates for quantum algorithms can be cast as a state generation problem. We define a paradigm for state generation, called 'adiabatic state generation' and develop tools for adiabatic state generation which include methods for implementing very general Hamiltonians and ways to guarantee non negligible spectral gaps. We use our tools to prove that adiabatic state generation is equivalent to state generation in the standard quantum computing model, and finally we show how to apply our techniques to generate interesting superpositions related to Markov chains.

1 Introduction

The paper reframes quantum algorithm design as quantum state generation, introducing an adiabatic framework with general Hamiltonian paths and tools for simulation and spectral-gap control. It proves equivalence with standard circuit-based state generation and applies the framework to SZK-related sampling and slowly varying Markov chains.

Motivation

  • The paper targets quantum state generation as an alternative framework for developing qualitatively different quantum algorithmic techniques.It studies states associated with classical probability distributions, called quantum sampling or Qsampling.Source: §1 · pp. 2–3

Quantum sampling and SZK

  • Most candidate quantum-algorithm problems considered in the paper reduce to quantum sampling, including DLOG, quadratic residuosity, lattice problems, and graph isomorphism.Theorem 1 states that every language in SZK reduces to circuit quantum sampling, implying that a general CQS solution would give SZK ⊆ BQP.Source: §1 · p. 3

Adiabatic state generation

  • Adiabatic state generation uses arbitrary paths of simulatable Hamiltonians rather than only local Hamiltonians on a straight line, while remaining efficiently reducible to quantum circuits.The framework is intended as a language for studying state generation, not as a replacement for the circuit model.Source: §1 · pp. 4–5

Hamiltonian simulation

  • Row-sparse, row-computable Hamiltonians with polynomial spectral norm are simulatable, providing a broad implementation tool for adiabatic state generation and quantum walks.The lemma also removes unnecessary node-coloring constraints in a cited quantum-walk implementation.Source: §1 · p. 5

Spectral-gap control

  • A jagged path through projections onto ground-state-orthogonal subspaces maintains non-negligible gaps when successive ground states have non-negligible overlap.Phase estimation implements the projections, and straight-line interpolation between suitable projections supplies the gap guarantee.Source: §1 · p. 6

Equivalence

  • Adiabatic state generation and standard circuit-model state generation are equivalent up to polynomial terms.Thus moving to the adiabatic framework and working with Hamiltonian ground states does not change the computational power of state generation.Source: §1 · p. 6

Markov-chain applications

  • The framework converts approximate-counting algorithms based on slowly varying strongly samplable Markov chains into efficient quantum sampling algorithms for final limiting distributions.It also yields simulatable Hamiltonians for strongly samplable chains and connects Markov-chain eigenvalue gaps with adiabatic spectral gaps.Source: §1 · pp. 6–7

Applications and connections

  • The methods Qsample perfect matchings, linear extensions, and restricted lattice points without requiring self-reducibility, linking quantum state generation to rapidly mixing Markov chains.The paper presents these applications as qualitatively different from previous state-generation techniques.Source: §1 · p. 7

2 Qsampling and SZK

This section develops Qsampling as a route from statistical zero knowledge to quantum algorithms, then illustrates reductions for discrete log, quadratic residuosity, and approximate CVP.

2.1 SZK

  • SZK contains promise problems characterized by distinguishing whether two circuit-output distributions are far apart or close in variation distance.The complete problem uses circuits C0 and C1 with the promise ||DC0 − DC1|| ≥ α or ||DC0 − DC1|| ≤ β.Source: §2.2 · pp. 9–10

2.3 Reduction from SZK to Qsampling.

  • Qsampling the corresponding circuits is sufficient to decide every problem in SZK with a quantum algorithm.The reduction prepares a superposition of the two sampled states, applies a Hadamard measurement, and amplifies the resulting constant gap.Source: §2.3 · pp. 10–11

2.4 A promise problem equivalent to Discrete Log

  • For discrete log, Qsampling a circuit computing y·g^i modulo p yields states whose supports can become disjoint under the promise, enabling the reduction.The construction uses the promise problem DLP1/6, which is related to discrete log and has a perfect zero knowledge proof.Source: §2.4 · pp. 11–12

2.5 Quadratic residuosity

  • For quadratic residuosity, the reduction maps r to xr^2 modulo n, producing identical sampled states for residues and separated behavior for non-residues.The argument assumes (x,n)=1 and uses the equal number of square roots in the n=pq case.Source: §2.5 · pp. 12–13

2.6 Approximating CVP

  • For the gap version of CVP, shifted noisy lattice samples have zero overlap when the target is far and non-negligible overlap when it is close.The close case yields fidelity at least n^-2c, and polynomial repetition gives a quantum algorithm with one-sided error.Source: §2.6 · pp. 13–14

3 Physics Background

This section introduces operator-norm and Hamiltonian-simulation tools, Schrödinger evolution, and the adiabatic theorem used to connect slowly varying Hamiltonians with ground-state generation.

3.1 Spectral Norm

  • The spectral norm measures the largest singular value, equals the largest absolute eigenvalue for Hermitian or unitary operators, and is submultiplicative.For a unitary U, ||U|| = 1.Source: §3.1 · p. 15

3.2 Trotter Formula

  • Trotterization approximates evolution under a sum of bounded Hamiltonians by interleaving evolutions under the individual terms.The approximation error after time t is bounded by O(MΛ · δ + MΛ^3t · δ^2), which vanishes as δ decreases.Source: §3.2 · p. 15

3.3 Time Dependent Schrodinger Equation

  • A quantum state evolves unitarily according to Schrödinger’s equation, with a time-independent Hamiltonian giving the familiar unitary evolution e^-iHT.The groundstate is the eigenstate of smallest eigenvalue, and the spectral gap separates it from the second-smallest eigenvalue.Source: §3.3 · p. 16

3.4 The adiabatic Theorem

  • The adiabatic theorem states that sufficiently slow evolution from a unique initial groundstate remains close to the final Hamiltonian’s groundstate under the stated regularity and gap condition.The theorem guarantees final distance at most ϵ^c for some constant c > 0.Source: §3.4 · pp. 16–17

4 Adiabatic Quantum State Generation

The paper defines adiabatic quantum state generation using general simulatable Hamiltonian paths and shows that this model can be efficiently simulated by standard quantum circuits.

4 Adiabatic Quantum State Generation

  • Adiabatic state generation permits general Hamiltonian paths, provided each Hamiltonian is efficiently simulatable by quantum circuits.The model requires differentiability except at finitely many points and circuit access to the initial groundstate and approximate short-time evolutions.Source: §4 · pp. 17–18
  • A simulatable Hamiltonian remains simulatable under polynomial scaling, convex combinations of bounded-norm simulatable Hamiltonians, and efficient unitary conjugation.These composition rules cannot be applied unboundedly many times because simulation complexity can blow up.Source: §4 · p. 17
  • Using a single total delay parameter T does not restrict the model because more complicated delay schedules can be encoded in the path’s dependence on s.This differs from earlier formulations that used a schedule τ(s).Source: §4 · p. 18

4.1 Circuit simulation of adiabatic state generation

  • Every adiabatic quantum state generator with polynomial runtime can be simulated by a quantum circuit that prepares the final groundstate to the required accuracy.The circuit discretizes the path and applies short-time Hamiltonian evolutions; an alternative proof uses repeated measurements and the Zeno effect.Source: §4.1 · p. 19

5 The Sparse Hamiltonian Lemma

The sparse Hamiltonian lemma reduces sparse, row-computable Hamiltonians to efficiently simulatable 2 × 2 combinatorial blocks, then combines their simulations to simulate the original Hamiltonian.

5 The Sparse Hamiltonian Lemma

  • A row-sparse, row-computable Hamiltonian with polynomial norm is simulatable.The proof decomposes it into polynomially many bounded-norm components and applies Trotter simulation.Source: §5 · p. 20, §5.3 · p. 23

5.1 The reduction to 2 × 2 combinatorially block diagonal matrices.

  • The decomposition colors H's entries so H = sum_m Hm, with every Hm row-computable and 2 × 2 combinatorially block diagonal.The coloring records modular, row-index, column-index, and entry-position information to isolate blocks.Source: §5.1 · pp. 20–21
  • Each colored component consists of combinatorial blocks of size at most 2 × 2, because the coloring prevents other entries from sharing the relevant rows or columns.The resulting blocks are either 1 × 1 or 2 × 2.Source: §5.1 · pp. 20–21, Figure 1 · p. 21
  • Every component satisfies ||Hm|| ≤ ||H||, preserving the polynomial norm bound needed for simulation.The component norm is attained on one of its 1 × 1 or 2 × 2 blocks and is bounded by the norm of H.Source: §5.1 · p. 21

5.2 2 × 2 combinatorially block diagonal matrices are simulatable.

  • A 2 × 2 combinatorially block diagonal Hamiltonian is simulatable by implementing the corresponding one- or two-dimensional unitary action on each block.Approximating each block's entries and evolution yields operator-norm error at most the target accuracy.Source: §5.2 · pp. 22–23

6 The Jagged Adiabatic Path Lemma

The jagged adiabatic path lemma replaces potentially gap-closing straight paths with paths through projections whose spectral gaps remain non-negligible, enabling efficient ground-state transport.

6 The Jagged Adiabatic Path Lemma

  • A sequence of simulatable Hamiltonians with non-negligible gaps and non-negligible neighboring ground-state overlap can be traversed efficiently from the first to the final ground state.The construction replaces each Hamiltonian by a projection onto the orthogonal complement of its ground state.Source: §6 · pp. 23–24
  • The resulting path satisfies the adiabatic condition under a linear time parameterization.The projections have bounded norm and unit gap, while the interpolation gaps are controlled by Claim 7.Source: §6 · pp. 23–24
  • A jagged path through a connecting Hamiltonian can keep the spectral gap above one, whereas the direct path's gap becomes zero.Figure 2 contrasts the gap profiles of the straight and jagged paths.Source: Figure 2 · p. 24
  • Hamiltonian-to-projection conversion is efficient when the Hamiltonian is simulatable, bounded, has groundvalue zero, and has gap larger than 1/n^c.Kitaev's phase estimation identifies the ground-state eigenspace, applies a conditional phase, and then uncomputes the eigenvalue information.Source: §6 · pp. 24–25
  • For projections onto two ground-state complements, every convex interpolation has gap at least the absolute ground-state overlap.Claim 7 reduces the gap analysis to the two-dimensional span of the two ground states.Source: §6 · p. 25

7 Equivalence of Standard and Adiabatic State Generation

The paper proves that standard quantum-circuit state generation and adiabatic state generation have equivalent efficient power by constructing an adiabatic path that follows a circuit's gate sequence.

7 Equivalence of Standard and Adiabatic State Generation

  • Every quantum circuit's final state can be produced by a quantum adiabatic state generator with complexity polynomial in the input size and circuit length.This is Claim 8 and supplies the circuit-to-adiabatic direction of the equivalence.Source: §7 · p. 25
  • The circuit is modified by replacing each gate with two square-root gates, making consecutive intermediate states sufficiently close.The modified circuit computes the same function as the original circuit.Source: §7 · p. 26
  • The adiabatic path linearly interpolates between Hamiltonians whose unique ground states are the successive states of the modified circuit.For s = j + η, Hx(s) = (1 − η)Hx(j) + ηHx(j + 1).Source: §7 · p. 26
  • Successive ground states have overlap at least 1/2, so Claim 7 guarantees non-negligible gaps along the interpolated path.The overlap bound is obtained by decomposing a state into eigenspaces of a square-root gate.Source: §7 · p. 26

8 Quantum State Generation and Markov Chains

The paper maps reversible Markov chains to Hamiltonians whose ground states encode limiting distributions, then uses adiabatic state generation to quantum-sample along suitable chains.

8.2 Reversible Markov chains and Hamiltonians

  • For a reversible Markov chain M, HM has norm at most 1, its spectral gap equals M's second eigenvalue gap, and its ground state is the square-root encoding of the limiting distribution.This establishes the correspondence between rapidly mixing chains and gapped Hamiltonians.Source: §8.2 · pp. 27–28
  • The Hamiltonian correspondence connects Hamiltonian simulation, spectral gaps, and ground states with rapidly mixing reversible Markov chains and their limiting distributions.The correspondence provides the basis for transferring adiabatic state-generation tools to quantum sampling.Source: §8.2 · pp. 27–28

8.3 Simulating HM

  • Strongly samplable Markov chains yield simulatable HM Hamiltonians, although strong samplability imposes a restrictive condition beyond row computability.The condition holds for Metropolis chains and trivially for symmetric chains.Source: §8.3 · p. 28

8.4 From Markov chains to Quantum Sampling

  • An efficient Qsampler for the initial distribution extends to the final distribution of a slowly varying sequence of strongly samplable, rapidly mixing Markov chains.Rapid mixing supplies spectral gaps, while slowly varying distributions ensure non-negligible neighboring ground-state overlap.Source: §8.4 · p. 29
  • The framework applies to superpositions associated with perfect matchings, spanning trees, lattice points in convex bodies, Gibbs distributions, and log-concave distributions.For perfect matchings, measuring the final state yields the uniform superposition over perfect matchings with non-negligible probability.Source: §8.4 · p. 29

8.5 Qsampling from Perfect Matchings

  • The approach is qualitatively different from simpler techniques because it does not rely on self-reducibility, though relaxing strong samplability could enable a graph-isomorphism application.The paper illustrates the method using Markov chains for perfect and near-perfect matchings.Source: §8.4 · p. 29, §8.5 · p. 30

A Zeno effect approach to simulating adiabatic Generators

The section replaces continuous adiabatic evolution with repeated ground-state measurements along a discretized Hamiltonian path. A spectral-gap-based overlap bound controls projection success, while phase estimation makes the measurements efficient and yields high-probability preparation of the final ground state.

Zeno discretization

  • Repeated projective measurements at R time steps follow the ground state from H(0) to H(1), with R chosen polynomially in the schedule parameters.The procedure starts in α(H(0)) and measures the ground-state subspace of H(j/R) at each step; R is polynomially related to the schedule time T.Source: A Zeno effect approach to simulating… · p. 34

Ground-state stability

  • If two Hamiltonians are η-close and each has spectral gap at least ∆, their ground states satisfy |⟨α(H)|α(J)⟩| ≥ 1 − 4η^2/∆^2.The proof compares the ground-state eigenvectors of H and J after shifting both Hamiltonians to be positive, then bounds their component outside the new ground state.Source: A Zeno effect approach to simulating… · pp. 34–35

Projection accuracy

Global error

  • The probability of making an error at any of the R measurement steps is at most O(Rη^2/∆^2), which is kept small by the choice of R.The supplied proof fragment states that the accumulated error over all steps is bounded by this order and that the relevant truncation parameter is chosen so the error is at most ǫ.Source: A Zeno effect approach to simulating… · p. 34

Efficient implementation

  • When H is simulatable and has a non-negligible spectral gap, phase estimation implements the required eigenvalue measurement efficiently with polynomially good confidence.Efficient simulation of e^−iHt for polynomial t enables the phase-estimation procedure, whose eigenstate-subspace outcome realizes the needed orthonormal measurement.Source: A Zeno effect approach to simulating… · p. 34
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