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A physical and universal model of bosonic computations with Solovay-Kitaev theorem

Dorian Rudolph, Arsalan Motamedi, Dhruva Sambrani, Hamid Reza Naeij, Ulysse Chabaud, Sevag Gharibian, Saeed Mehraban

arXiv:2609.40226v1quant-phcs.CC

TL;DR

Existing bosonic models can permit unbounded energy, which the paper connects to unreasonable computational power and failures of robust compilation. The paper introduces BEQC, where energy is an input resource and gates preserve photon number. It shows that polynomial-energy BEQC recovers BQP and supports universal energy-preserving compilation.

  • Problem

    The Lloyd–Braunstein model permits unbounded energy, which can yield unreasonable computational power and undermine universal, efficient compilation.

    Source: §1 · p. 4

  • Method

    BEQC supplies energy through an input coherent state and restricts gates to unitaries that preserve photon number.

    Source: §1.1 · p. 5, §1.1.1 · p. 5

  • Results

    BEQP with polynomial time, polynomial energy, and constant precision equals BQP.

    Source: §1.1.2 · p. 6

  • Takeaways & Limitations

    BEQC combines input-controlled energy with a universal gate set and efficient Solovay–Kitaev compilation for energy-preserving unitaries.

    Source: §1.1 · p. 5, §1.1.2 · p. 8

    The stated robustness to imperfect resource states applies to a number of simulation results, rather than being stated for every result.

    Source: §1.1.1 · p. 6

Abstract

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Bosonic quantum systems are among the leading architectures for quantum information processing, offering continuous-variable degrees of freedom with strong error-correction capabilities. However, standard bosonic quantum computation models such as the Lloyd-Braunstein [Lloyd and Braunstein, 1999] and hybrid oscillator-qubit models [Brenner, Dias, and Koenig, 2025; Liu et al., 2026] permit dramatic energy growth, leading to unphysical computational power and the breakdown of fundamental algorithmic tools such as universal and efficient compilation [Brenner et al., 2026; Rudolph et al., 2025]. To address this, we introduce a new model of bosonic quantum computation, Bosonic Energy-Preserving Quantum Computation (BEQC), in which energy is treated as a computational resource. Namely, energy is supplied solely through input coherent states, and all gates are generated by energy-preserving Hamiltonians. Thus, by construction, dramatic energy growth is impossible, making the model physically grounded. We next show that BEQC is a computationally robust and universal model in many respects, including: (1. Computational power) BEQC efficiently simulates all polynomial-energy computations in existing models, and exactly recovers BQP in the polynomial energy setting. It further admits several complexity-theoretic upper bounds when varying the energy, precision, and space parameters of the model. (2. Universal gate sets and state synthesis) BEQC has natural universal gate sets based on linear optics and Kerr interactions. In particular, we obtain a Solovay-Kitaev theorem which circumvents previous no-go results. We give various applications, including (a) a protocol for engineering GKP states with rigorous preparation guarantees, (b) Fock state preparation to exponential precision, and (c) native Fock space simulation of any qubit-based unitary.

1 Introduction

The paper argues that unbounded energy undermines the physical grounding and algorithmic robustness of bosonic computation, then introduces BEQC to control energy as an input resource. It reports polynomial-energy simulation, recovery of BQP, universal energy-preserving gate sets, and applications to compilation and state preparation.

1 Introduction

  • In the Lloyd–Braunstein model, unbounded energy can yield unreasonable computational power, while unbounded operators undermine Trotter splitting and strong Solovay–Kitaev compilation.The cited example shows O(k)-mode gates can efficiently solve NTIME(exp(k)); the paper also gives a counterexample to efficient compilation claims for Gaussian and cubic-phase gates.Source: §1 · p. 4

1.1.1 Defining the model, BEQC: Energy from a resource state

  • BEQC confines energy growth by using only photon-number-preserving gates and supplying computational energy through an input coherent state.The coherent state's expected photon number is |α|^2, so the input parameter α controls the model's energy.Source: §1.1.1 · p. 5

1.1.2 Computational properties

  • BEQP with polynomial time, polynomial energy, and constant precision equals BQP; changing energy, precision, space, locality, or Hamiltonian degree yields further complexity bounds.Logarithmic energy with logarithmically many modes gives a class in P, while non-local polynomial-degree Hamiltonians make BEQP PSPACE-complete.Source: §1.1.2 · pp. 6–7
  • BEQC efficiently simulates computations in several models when their energy remains polynomially bounded, including Lloyd–Braunstein, hybrid CV-DV, qubit, fusion-based, and fermionic computation.For polynomial Hamiltonians, the parametric-approximation result requires a finite energy-moment bound; choosing sufficiently large input energy gives arbitrarily good approximation.Source: §1.1.2 · pp. 7–8
  • Energy-preserving linear optics with either self-Kerr or cross-Kerr gates forms a universal set with an energy-preserving Solovay–Kitaev theorem.For constant m or K, circuits use poly(m, K, log(1/ε)) gates; the formal theorem also gives precision dependence log^c_SK(1/ε), with c_SK < 4.Source: §1.1.2 · p. 8
  • The compilation results also provide a route to simulating qubit-described unitaries in Fock space using a unary encoding of Fock states.The approach first applies an isometry flattening Fock states into unary-encoded qubit states, then applies the qubit circuit.Source: §1.1.2 · pp. 8–9
  • BEQC compiles non-local number-preserving unitaries and local non-number-preserving unitaries from efficient qubit descriptions, with different resource costs.For non-number-preserving targets, the stated compilation uses energy polynomial in the desired precision rather than logarithmic in it, and may require gates beyond the Kerr-based sets.Source: §1.1.2 · pp. 8–9
  • The Solovay–Kitaev theorem yields deterministic preparation of finite Fock-state superpositions and high-quality GKP states within inverse-polynomial trace distance, without intermediate measurements or postselection.The paper also reports robustness to thermal input states or perturbations of the coherent resource state.Source: §1.1.2 · p. 6, e. State synthesis · p. 9

e. State synthesis.

BEQC supports rigorous state-synthesis and Bose–Hubbard simulation results, while its energy-preserving structure motivates further questions about computational power, compilation, and practical architectures.

e. State synthesis.

  • Any m-mode state with at most K photons can be synthesized using m + 2 modes with trace distance at most ε and resources polynomial in the stated parameters.The theorem gives gate-count and classical-compilation bounds involving poly(m, K, 1/ε)poly(K, 1/ε)^O(m).Source: e. State synthesis · p. 9
  • A deterministic three-mode circuit prepares GKP states within trace distance ε using one coherent input, two vacuum inputs, and linear optics with self-Kerr or cross-Kerr gates.Its gate count is polynomial in κ^-1, Δ^-1, and ε^-1.Source: e. State synthesis · p. 9
  • Energy-preserving Bose–Hubbard dynamics can be decomposed efficiently on bounded-particle sectors and extended to bounded expected photon number.For coherently driven dynamics, a coherent-state ancilla gives an explicit tradeoff between supplied energy and simulation error.Source: e. State synthesis · pp. 9–10

1.2 Open questions

  • Energy is controlled by construction, and the Solovay–Kitaev theorem offers controlled-accuracy compilation, while fault tolerance and Kerr-gate injection remain practical questions.The paper identifies experimental implementation and whether nonlinear gates can be injected using linear optics and measurements on resource states as open directions.Source: §1.2 · p. 10
  • Whether ballistic energy-preserving bosonic evolution with explicit resource states matches gate-based computational power, and whether homodyne measurements change that power, remain open.The homodyne question specifically concerns whether phase-sensitive measurements distinguish pure coherent resources from thermal resources without phase information.Source: §1.2 · p. 11
  • Open problems include tradeoffs among depth, energy, modes, ancillas, and noise, as well as bosonic Hamiltonian complexity and finite-temperature state or observable approximation.The stated finite-temperature questions include approximating Gibbs states, partition functions, and thermodynamic observables for restricted bosonic Hamiltonians.Source: §1.2 · p. 11
  • The energy-preserving Solovay–Kitaev theorem currently targets strong-cutoff approximation and the Kerr-plus-linear-optical gate set; naive smooth-cutoff truncation has polynomial dependence on 1/ε.For states with bounded exponential energy, truncation can instead scale as O(E + log(1/ε)); the passage also identifies characterizing such gate sets as an open question.Source: §1.2 · p. 12
  • Further directions include connecting BEQC to resource theories, extending exponential-energy fast-forwarding beyond one oscillator, and characterizing robustness under model variations.The paper also raises the question of whether the model's computational power is unchanged in a ballistic formulation.Source: §1.2 · pp. 10–12

2 Preliminaries

This section defines bosonic states and gates and introduces energy-sensitive norms and bounds for analyzing operations and approximations, including how operators change photon number on a cutoff subspace.

2.1 Definitions

  • Bosonic modes use ladder operators and Fock states, with total photon number defining a state’s energy as its expectation.Multimode systems are tensor products of single-mode spaces.Source: §2.1 · pp. 13–14
  • Energy-preserving Hamiltonians commute with total photon number, equivalently having equal creation and annihilation counts in every normal-ordered monomial.Linear optics, Kerr, and cross-Kerr gates are energy-preserving examples.Source: §2.1 · pp. 15, 17
  • Gaussian states arise from thermal states through degree-at-most-2 Hamiltonians; Gaussian gates preserve them and Schwartz space, while higher-degree preservation is undecidable in general.The gates used in this work are chosen to preserve Schwartz space.Source: §2.1 · pp. 16–17

2.2 Energy-constrained norms

  • The energy-constrained diamond norm tests channel differences on states with expected photon number at most E and avoids standard diamond-norm pathologies in infinite dimensions.An energy moment bound controls energy growth during polynomial-Hamiltonian evolution and appears in simulation error bounds; a lemma relates cutoff norm to this energy-constrained norm for K > E.Source: §2.2 · p. 18
  • For cutoff K, r annihilation operators leave at most K−r photons and vanish when r>K; r creation operators can add at most r photons.These photon-number bounds establish the first claim of the argument.Source: §2.2 · pp. 19–20
  • The bound for A = λMα,β + λM† is obtained by applying the same estimate to M†.The supplied passage does not include the bound’s explicit value.Source: §2.2 · p. 20

3 Why energy-preserving gates? The issues of active gate decompositions

This section examines why active bosonic gate decompositions can fail operator-norm control and produce extreme output energies, motivating energy-preserving operations supplied with energy resources. It constructs an active-gate evolution whose output position exceeds a threshold with high probability and uses a recurrence argument to establish the tail bound.

3 Why energy-preserving gates? The issues of active gate decompositions

  • Energy-preserving Hamiltonians decompose into finite-dimensional operators on each N-photon subspace, enabling efficient cutoff compilation and simulation of energy-bounded non-energy-preserving circuits with resource states.The paper states that cutoff compilation uses linear optics and Kerr gates, while resource states supply energy for simulating non-energy-preserving circuits under energy-moment bounds.Source: §3 · p. 20

3.1 No convergence in operator norm

  • For linear-optical Hamiltonians, Trotter error can approach 2—the maximum operator-norm distance between unitaries—even though the product formula converges in the strong operator topology.The example preserves photon number, so the product formula converges strongly while failing to converge in operator norm.Source: §3.1 · p. 21

3.2 Doubly exponential energy despite convergence

  • Trotter splitting for A = q^2 + p^2 and B = q^4 converges strongly, yet vacuum-input energy diverges doubly exponentially with the number of steps while exact evolution keeps it O(1).The exact-evolution energy bound holds for all t.Source: §3.2 · p. 24

3.3 A no-go for efficient approximate synthesis of active gates

  • A bounded-runtime family of n number-controlled squeezing and Gaussian gates produces at least 2 ⇈Ω(n) photons with probability at least 1−2^-Ω(n), while even exp(n) Gaussian/cubic gates yield exponentially small output overlap.This obstructs efficient approximate synthesis of the family without a priori energy bounds.Source: §3.3 · pp. 25–26
  • An O(n^2)-gate bounded-time Gaussian/cubic circuit can put all but e^-Ω(n^2) probability above 2^(2n−O(1)) photons, while 2^o(n) Gaussian/Kerr gates achieve exponentially small output overlap.The high-occupation probability bound is stated for the output mode on vacuum input.Source: §3.3 · p. 26
  • The input preparation uses O(n) constant-strength squeezing gates per mode, for O(n^2) gates total, and displaces the first control mode.The output position variables follow the recurrence Y_(j+1) = Q_(j+1) + Y_j^2.Source: §3.3 · pp. 27–28
  • The output position satisfies p_n ≥ 1 - e^(-n^(2/3)) for sufficiently large n.Here p_n is the probability that the measured output position has absolute value at least R_n.Source: §3.3 · p. 28
  • On event E_n, the recurrence yields Y_n ≥ 7/8 μ_n = R_n, establishing the position threshold used in the tail bound.The proof tracks relative error inductively across the recurrence Y_(j+1) = Q_(j+1) + Y_j^2.Source: §3.3 · pp. 28–29

4 Bosonic Energy-Preserving Quantum Computation (BEQC) and its BQP analogue, BEQP

BEQC defines bosonic circuits whose energy is bounded by the energy of their coherent-state input, with gates generated by energy-preserving polynomial Hamiltonians. BEQP is the corresponding promise class, with a polynomial-energy version and universal gate sets satisfying a Solovay–Kitaev theorem.

4.1 Bosonic Energy-Preserving Quantum Computation (BEQC)

  • BEQC bounds circuit energy by the coherent-state input energy and uses gates generated by energy-preserving polynomial Hamiltonians.The model also specifies vacuum modes, a designated output mode measured by photon number, and circuit depth.Source: §4.1 · p. 30
  • Equal-total-energy coherent states over arbitrary mode counts are efficiently interconvertible using linear optics.The construction pads with vacuum modes when the input and output mode counts differ.Source: §4.1 · p. 31

4.2 The complexity class BEQP

  • BEQP defines promise problems using polynomial-time uniform BEQC circuits, polynomial runtime, and efficiently computable completeness and soundness bounds.With polynomial energy, the class is denoted BEQP; repetition can improve completeness and soundness, but requires polynomially increased energy.Source: §4.2 · p. 31
  • BEQP has universal Kerr and linear-optics gate sets that obey a Solovay–Kitaev theorem.The paper consequently drops the specific gate set when assuming universality.Source: §4.2 · p. 31

5 Parametric approximation of active gates by energy-preserving gates

This section shows how BEQC uses ancillary coherent states and energy-preserving operations to approximate active dynamics, with guarantees that include robustness to noisy resources. It also describes simulations of polynomial-Hamiltonian and established computation models, including measurement-based computation, and applications to GKP-state preparation and fermionic computation.

5.1 Parametric approximation

  • BEQC replaces active Hamiltonian terms with energy-preserving interactions involving an ancillary coherent state, achieving O(1/α) error when the simulated evolution’s energy is controlled.The construction adds ancilla creation or annihilation factors to Wick-ordered monomials; a displacement Hamiltonian, for example, becomes a beamsplitter interaction.Source: §5.1 · p. 32, §5 · p. 32

5.2 Robust parametric approximation

  • Displaced low-energy resource states can replace coherent ancillas while keeping simulated evolution close to the ideal output, including for thermal noise under an additional displacement condition.Finite mean photon number alone does not guarantee error linear in time for arbitrary resource states; the thermal-state result has that time dependence under its stated assumptions.Source: §5.2 · pp. 36–38

5.3 Simulating existing models within BEQC

  • Under polynomial energy-moment bounds, BEQC simulates DV, CV, and hybrid computations with polynomial overhead, including cubic gates and exactly energy-preserving conditional rotations.The section also gives a route under a polynomial mean-energy bound when finite-cutoff gate approximations have efficiently constructible qubit-circuit descriptions.Source: §5.3 · pp. 38–39
  • Under Theorem 5.2’s assumptions, polynomial-Hamiltonian gates are covered, CV-DV interactions use dual-rail encodings, and the construction also covers KLM.More general gates can be treated using Theorem 6.22 when their finite-cutoff versions have efficient qubit-circuit descriptions.Source: §5.3 · p. 39
  • BEQC simulates fusion-based quantum computation with polynomial overhead, replacing intermediate measurements and feed-forward by retained detector modes and a final number measurement.Constant-sized resource states use dual-rail gates and approximate single-photon preparation; fusion measurements use linear optics and photon counting, while reversible classical processing controls subsequent operations.Source: §5.3 · pp. 39–40
  • BEQC can prepare finite-energy approximate GKP states of high quality with a polynomial-size circuit and a polynomial-energy coherent-state ancilla.The passage points to Section 6.3.2 for the preparation demonstration.Source: §5.3 · p. 40
  • Fermionic computations can be reproduced by simulating qubit computations through the Jordan–Wigner mapping.Source: §5.3 · p. 40

6 Circuit compilation and state synthesis

This section develops energy-preserving compilation methods for qubit-described and Fock-space unitaries, including Solovay–Kitaev bounds, coarse compilation, and encoding gadgets. It also presents deterministic state-synthesis methods, including approximate GKP-state preparation with polynomial resource bounds.

6.1.1 Setup and main theorem

  • Finite subsets of the self-Kerr and cross-Kerr gate sets retain the theorem’s guarantees independently of cutoff and precision.The construction uses linear optics together with either Kerr gate set.Source: §6.1.1 · p. 41

6.1 Efficient Solovay–Kitaev compilation with linear optics and Kerr gates

  • Energy-preserving unitaries admit deterministic Solovay–Kitaev compilation with linear-optical and Kerr gates in poly(m, K, D, log(1/ε)) bit operations.The theorem approximates the unitary on the cutoff space; its error is measured using the cutoff operator seminorm.Source: §6.1 · p. 40, §6.1.1 · p. 41

6.1.2 On the ancilla

  • An extra vacuum ancilla enables compilation of arbitrary photon-sector determinant patterns, while sectorwise determinant-one unitaries can be compiled without ancillas.Without the ancilla, phase patterns across photon-number blocks are constrained by two gate parameters; the vacuum ancilla also creates a complication for recursive refinement because it may become entangled with the data.Source: §6.1.2 · pp. 41–42

6.1.3 Proof outline

  • The proof decomposes sectorwise determinant-one unitaries into O(D²) adjacent-Fock-state two-level operations, builds a coarse compiler, then recursively refines its error.The adjacent states differ by moving one photon between modes; the coarse compiler must act on the full cutoff space without entangling the environment.Source: §6.1.3 · p. 43

6.1.4 Rotations on adjacent Fock states

  • Adjacent Fock-state SU(2) control uses quadratic number-operator Hamiltonians and finite Fourier cancellation to isolate a desired hopping transition, with Trotter errors bounded rigorously.The construction cancels unwanted transitions exactly within the cutoff space before accounting for errors from implementing the Hamiltonian evolution.Source: §6.1.4 · pp. 43, 44, 47
  • The construction produces energy-preserving circuits; each Qr uses O(LJ) = O(m²Λ²/ξ) = O(m⁶q²K⁴/ξ) gates, while the triangle inequality gives ∥eC −eitP∥(K) ≤ η/3.Source: §6.1.4 · p. 48

6.1.5 Coarse compiler

  • For U∈SU(D0,…,DK), the coarse compiler is polynomial-size at fixed ε0 and computable in poly(m,K,D,ε0^-1) time.The construction decomposes sector unitaries into adjacent two-level operations before compiling them.Source: §6.1.5 · pp. 49–50

6.1.6 Solovay–Kitaev details

  • With a universal gate set containing inverses and a coarse compiler, block-diagonal SU targets admit ε-approximations using O(L(ε0) log^c_SK(1/ε)) gates when ε0≤min{10^-5,1/d_j}.The theorem assumes an (L,T)-coarse compiler and applies to U∈SU(d1,…,dK).Source: §6.1.6 · p. 50
  • For δ≤1/d, Lemma 6.13 computes b-bit approximations to the commutator factors in d^4 poly(b+log d) bit operations.The input matrix Δ must be supplied to 2b+O(log d) bits of precision.Source: §6.1.6 · p. 54
  • The construction returns entries to b bits of precision with operator-norm error O(d2^-b), using d4poly(b + log d) bit operations.Finite-precision diagonalization of eH is the dominant step.Source: §6.1.6 · p. 57

The computation can be performed using

  • A near-identity determinant-one unitary admits a computable traceless Hermitian logarithm approximation, with finite-precision construction taking md3poly(p + log d + log m) bit operations.The computation uses a truncated logarithm and finite-precision arithmetic.Source: §6.1.6 · p. 58, The computation can be performed using · p. 60

6.2 Compiling general unitaries in Fock space

  • The framework compiles energy-preserving unitaries on constant numbers of modes and extends to Fock-space unitaries with efficient qubit circuit descriptions, including number-preserving and non-number-preserving cases.Parametric approximation also enables compilation of constant-degree polynomial-Hamiltonian gates under the energy-moment assumptions of Theorem 5.2.Source: §6.2 · p. 61

6.2.1 Encoding gadgets

  • Energy-preserving gadgets implement Fock-to-unary and unary-to-dual-rail encodings, providing tools to import qubit-based results into the bosonic setting.The unary-to-dual-rail constructions use either an auxiliary Fock state or a coherent state.Source: §6.2.1 · pp. 62–63

6.2.2 Compiling number-preserving unitaries

  • Theorem 6.20 addresses compiling number-preserving unitaries on Hm,K from δ-approximate qubit circuits with L gates over q ≥ mK qubits and δ ≤ ε/4.It asserts the existence of a bosonic unitary circuit CU and a real α.Source: §6.2.2 · p. 64
  • Number-preserving unitaries with efficient qubit circuit descriptions compile with poly(m, K, L, log(1/ε)) gates and coherent-state energy O(mK + L + log(1/ε)).This includes Bose–Hubbard dynamics at total photon number K, compilable to exponential precision with a polynomial-size circuit; ancillary Fock states can be prepared using coherent-state energy O(q + log(1/η)).Source: §6.2.2 · pp. 64, 66

6.2.3 Compiling non-number-preserving unitaries

  • General qubit-described unitaries, including non-number-preserving ones, compile with polynomially many local constant-degree energy-preserving gates and a coherent input of energy O(q^2/ε^2).The input circuit must be a δ-approximate description with δ ≤ ε/4; the method extends BEQC simulation to computations whose mean energy, rather than all energy moments, stays polynomially bounded.Source: §6.2.3 · pp. 67–68

6.3 State synthesis

  • State synthesis prepares finite Fock superpositions and high-quality GKP states deterministically to inverse-polynomial trace-distance accuracy, using GKerr or G×Kerr gates without intermediate measurements or postselection.The passage contrasts this with usual GKP preparation protocols.Source: §6.3 · p. 69

6.3.1 Preparing finite Fock superpositions

  • For fixed m, any state with total photon number at most K can be prepared within trace distance ε using polynomially many linear-optical and Kerr or cross-Kerr gates.The construction uses m+2 modes, a single coherent input with the remaining inputs vacuum, and polynomial input energy and gate count in K and ε^-1.Source: §6.3.1 · pp. 69–70

6.3.2 Preparing Gottesman–Kitaev–Preskill states

  • The GKP-state analysis establishes an exponential photon-number-tail bound with R = κ^-2 + Δ^-2, supporting the section’s efficient GKP-state preparation result.The supplied passage states that a universal constant C ≥ 1 exists for the bound but does not include its full expression.Source: §6.3 · p. 69, §6.3.2 · p. 71
  • A three-mode circuit with one coherent and two vacuum inputs deterministically prepares a state within trace distance ε of the target GKP state using linear optics and self-Kerr or cross-Kerr gates.Its gate count is polynomial in κ^-1, Δ^-1, and ε^-1.Source: §6.3.2 · p. 72
  • The coherent energy is O(K^2/ε^2), yielding a gate bound O(poly(K_joint polylog(1/ε))) for K_joint = O(K^2/ε^2).Source: §6.3.2 · p. 72
  • Classical state specification takes O(JK) arithmetic steps using a recurrence for Fock coefficients, with polynomial guard bits when κ and Δ are rational or efficiently computable.Source: §6.3.2 · p. 72
  • Unlike, the protocol avoids a coherent implementation of ⌊q̂⌋, but its circuit depth is polynomial rather than logarithmic.Source: §6.3.2 · p. 72

7 Complexity of energy-preserving bosonic computations

This section characterizes the complexity of energy-preserving bosonic computations across coherent and thermal inputs, establishing BQP-completeness in the stated polynomial-energy and temperature-difference settings and identifying the simplicity of equal-temperature thermal inputs. It also establishes PSPACE-completeness for computations with non-local polynomial-degree energy-preserving gates, gives classical upper bounds varying with energy, mode count, and precision, and identifies a conditional barrier to efficiently compiling non-local gates into local ones.

7.1 BQP-completeness for coherent states

  • BEQP with polynomial-energy coherent inputs equals BQP.BQP simulations use polynomially many modes and polynomial energy; the reverse containment follows from efficient computation of energy-preserving gates’ Fock-basis entries.Source: §7.1 · pp. 73–74

7.2 BQP-completeness for displaced thermal states

  • N displaced thermal ancillas and one vacuum mode can yield a displaced thermal state with the same amplitude and thermal photon number reduced by a factor of N, enabling BQP-completeness with polynomial thermal noise.The resulting state is ρ(α, n̄/N); polynomially many noisy copies suffice for the BQP simulation.Source: §7.2 · pp. 74–75

7.3.2 Energy difference is enough

  • Thermal inputs at two temperatures with polynomially separated mean photon numbers support BQP-complete computation using polynomial-degree energy-preserving Hamiltonians.The construction encodes logical qubits in pairs of cold and hot thermal modes; the section notes a trade-off with error correction for constant-degree gates and constant energy separation.Source: §7.3 · p. 75, §7.3.2 · pp. 76, 79

7.3.1 Energy alone is not enough

  • When all thermal inputs have the same mean photon number, energy-preserving operations leave the input invariant, making the model computationally simple.The equal-temperature inputs form a Gibbs state of total photon number, so a single mode’s photon number can be computed from its thermal distribution.Source: §7.3.1 · p. 75

7.3.2 Energy difference is enough

  • If the hot and cold thermal mean photon numbers differ by poly(N), ThermalBEQPpoly with those temperatures is BQP-complete.The result also holds with constant-degree gates from GKerr (or G×Kerr) at polynomial overhead.Source: §7.3.2 · p. 79

7.3.3 Thermal energy-preserving with constant per-mode energy is BQP-complete

  • Routing fresh qubits confines failures to two rounds, while stochastic domination converts correlated physical errors into independent virtual gate errors that fault tolerance can correct below threshold.The virtual gate failure rate is pgate = 1 − (1 − √p)^∆, and the stated condition is pgate < pth.Source: §7.3.3 · pp. 80–83
  • ThermalBEQPpoly with constant mean photon numbers for both temperatures is BQP-complete, using polynomially many pairs and gates with constant per-mode energy and gate degree.The construction combines fault-tolerant error correction with a readout whose error is bounded to preserve the BQP promise.Source: §7.3.3 · p. 84

7.4 Computations with non-local energy-preserving gates of polynomial degree are PSPACE-complete

  • The PSPACE-completeness result implies that nonlocal unitaries cannot be efficiently decomposed into constant-local bosonic gates unless PSPACE = BQP.Circuits with polynomially many constant-local gates of polynomial degree can be simulated in BQP.Source: §7.4 · p. 84
  • Polynomially described nonlocal polynomial-degree energy-preserving gates yield PSPACE-complete computation, even with vacuum and coherent states as the only inputs.The input has polynomial expected photon number; the gate set may include a single evolution generated by a sum of constant-local Hamiltonians.Source: §7.4 · p. 84
  • The coherent-input simulation problem for non-local polynomial-degree energy-preserving gates is PSPACE-complete: it is PSPACE-hard and lies in PSPACE.The hardness construction uses coherent time averaging to simulate an exponentially long, polynomial-space computation; the upper bound uses an explicit classical polynomial-space simulation.Source: §7.4 · p. 87
  • Unless BQP = PSPACE, non-local unitaries generated by k-local polynomial-degree energy-preserving Hamiltonians cannot be efficiently compiled into local polynomial-degree gates to constant additive precision, even on polynomial-energy states.The text also notes that a related PSPACE-hardness result for constant-degree non-local operations at exponential energy remains open under the stated upper-bound condition.Source: §7.4 · p. 87

7.5 Bounding the complexity of energy-preserving computations

  • BEQP with O(1) modes and PreciseBEQP on O(1) modes are in P, while BEQP(log) with O(log) modes is in P.The theorem also states that BEQP with O(log) modes is in QP ∩ BQP.Source: §7.5 · p. 90
  • With polynomial energy, BEQP = BQP.This is the theorem’s exact characterization for the polynomial-energy setting.Source: §7.5 · p. 90
  • PreciseBEQP with polynomially many modes is contained in PSPACE.The proof uses an inverse-exponential precision cutoff and the polynomial-space simulation of Theorem 7.29.Source: §7.5 · p. 91

B Finite gate sets

A finite subset of the Kerr gate families suffices independently of cutoff and precision. Replacing tunable gates with fixed gates preserves polylogarithmic precision dependence in the coarse compiler.

B Finite gate sets

  • A finite subset of G_Kerr or G_×Kerr suffices independently of cutoff and precision.The model allows tunable evolution times, so the gate families in Eq. (121) are admissible as stated.Source: B Finite gate sets · p. 100
  • The selected generators have integer spectra, making their rotations 2π-periodic.The generators are 1, n̂₁n̂₂, and −i(â†₁â₂ − â†₂â₁), together with e^(2πiφH) and its inverse.Source: B Finite gate sets · p. 100
  • Any tunable elementary gate can be approximated to cutoff error δ using O(K²/δ) fixed gates.This follows from bounded partial quotients of φ = [0; 1, 1, ...] and cutoff generator norms O(K²).Source: B Finite gate sets · p. 100
  • Replacing tunable gates only in the coarse compiler before Solovay–Kitaev refinement retains polylog(1/ε) precision dependence.The replacement uses fixed gates; the refinement step preserves the stated precision dependence.Source: B Finite gate sets · p. 100

C Heisenberg picture with bosonic gates

This appendix justifies Heisenberg-picture calculations for bosonic gates despite unbounded operators. It establishes conditions under which the relevant dynamics and conjugation identities hold on Schwartz space.

C Heisenberg picture with bosonic gates

  • The appendix justifies Heisenberg-picture calculations for non-Gaussian gates, for which unbounded operators make the Baker–Campbell–Hausdorff formula unsafe to apply without proof.This also makes the energy bounds for Gaussian plus q̂³ circuits rigorous and justifies corresponding calculations in [KQ21].Source: C Heisenberg picture with bosonic gates · p. 100
  • The proof represents commutators in a finite-dimensional span and derives their evolution from a finite-dimensional coefficient matrix.The span contains A, and the matrix M encodes the commutators with H.Source: C Heisenberg picture with bosonic gates · p. 100
  • The derivation verifies that the Heisenberg operator satisfies the generalized equation of motion and the initial condition on Schwartz space.It then applies Corollary C.8 to identify the Heisenberg operator with A_H(t).Source: C Heisenberg picture with bosonic gates · p. 101
  • The evolution preserves Schwartz space, and the continuity step completes the associated argument.Claim C.2 states e^(itH)S = S; the proof then reduces to continuity, which Claim C.3 establishes.Source: C Heisenberg picture with bosonic gates · pp. 101–102
  • Under Theorem C.1’s conditions, conjugating e^(itA) by e^(isH) gives the evolution generated by A_H(s).The proof establishes the identity first on Schwartz space and extends it to all of L² by density.Source: C Heisenberg picture with bosonic gates · p. 102
  • Theorem C.1 covers polynomial observables with polynomial generators, including mixed position- and momentum-based generators.The mixed-generator result includes H = R(q̂₁, p̂₁, ..., q̂ₘ, p̂ₘ) with real polynomial R and one q̂ or p̂ operator per mode.Source: C Heisenberg picture with bosonic gates · p. 102

C.1 Uniqueness of the Generalized Heisenberg Equation

This section specifies the generalized Heisenberg equation and an admissible solution class. Under its stated operator and continuity assumptions, the Heisenberg operator is the unique solution on Schwartz space.

C.1 Uniqueness of the Generalized Heisenberg Equation

  • The generalized Heisenberg equation is defined through differentiability of matrix elements between vectors in a dense subspace.The operators must have dense domains containing that subspace, along with the domains of their adjoints and H.Source: §C.1 · p. 103
  • The admissible solution class assumes e^(itH)D = D and requires one of two pairs of strong-continuity conditions.The alternatives concern continuity of X(t)ψ and X(t)*e^(itH)ψ, or of X(t)e^(itH)ψ and X(t)*ψ.Source: §C.1 · p. 103
  • Under the stated assumptions, X(t) = A_H(t) is the unique solution on Schwartz space with initial condition A.H is self-adjoint, its evolution preserves Schwartz space, and the operators’ restrictions there are polynomials in q̂ and p̂.Source: §C.1 · p. 104

D Effective energy-preserving descriptions of energy-preserving operations

Theorem D.1 gives a constructive polynomial description of any finite-dimensional energy-preserving operation on a truncated bosonic space. The construction has bounded degree and can be computed efficiently in the input operator’s size.

D Effective energy-preserving descriptions of energy-preserving operations

  • Any Hermitian energy-preserving operator on the truncated subspace has an energy-preserving polynomial Hamiltonian description.The theorem assumes [Ĥ, N̂] = 0 on the subspace; the appendix presents this as a constructive description of energy-preserving operations.Source: D Effective energy-preserving… · p. 104
  • The constructed polynomial Hamiltonian has degree at most 6mN_max.Here m is the number of modes and N_max the local dimension cutoff.Source: D Effective energy-preserving… · p. 104
  • The construction uses local number projectors to restrict transition operators to the desired Fock-state transitions.Sandwiching bare ladder operators between projectors suppresses spurious transitions from higher-excited states.Source: D Effective energy-preserving… · p. 105
  • Summing the transition terms yields a global polynomial Hamiltonian that agrees with Ĥ on the truncated subspace.Outside that subspace, the constructed operator acts as some other operator Ĥ′.Source: D Effective energy-preserving… · pp. 105–106
  • The polynomial description can be computed in time polynomial in the size of Ĥ.The stated costs are O(N_max^(4m)) for dense Ĥ and sN_max^(2m) when Ĥ has s nonzero entries.Source: D Effective energy-preserving… · pp. 104, 106

E Evaluating Fock entries of energy-preserving Hamiltonians

Energy preservation makes the Hamiltonian block diagonal by total photon number, so its Fock-space entries can be computed through finite-dimensional photon-number blocks. The appendix bounds the precision and resources needed to approximate the block matrix exponential.

E Evaluating Fock entries of energy-preserving Hamiltonians

  • The squaring parameter s grows as O(1 + log(1 + |t|) + log(C_max) + log T_m,d + d log(1 + n)).Here T_m,d is the number of energy-preserving monomials; for fixed m, d, and C_max, the section gives a simplified bound.Source: E Evaluating Fock entries of… · p. 106
  • Energy-preserving monomials preserve total photon number, making the Hamiltonian block diagonal across photon-number sectors.The vacuum block is scalar and handled separately.Source: E Evaluating Fock entries of… · pp. 106–107
  • 2^-p-additive entrywise accuracy is achieved using numbers with O(p + s + log D_n + log T_m,d + log(p + s)) bits.The bound includes rounding after each squaring.Source: E Evaluating Fock entries of… · p. 109

F Application: gate decomposition for Bose–Hubbard dynamics

The appendix applies BEQC to Bose–Hubbard dynamics, treating both the standard number-preserving Hamiltonian and a coherently driven variant. It derives explicit simulation guarantees and compares the construction with related bosonic approaches.

F Application: gate decomposition for Bose–Hubbard dynamics

  • The standard Bose–Hubbard simulation uses an energy-preserving Lie–Trotter decomposition with explicit error and resource bounds on finite photon-number sectors.The result is extended to states with bounded mean photon number.Source: F Application: gate decomposition for… · p. 109
  • A coherent-state ancilla converts the coherently driven, non-number-preserving Bose–Hubbard dynamics into an energy-preserving simulation.The appendix also compares this construction with related bosonic simulation approaches.Source: F Application: gate decomposition for… · p. 109

F.1 Standard Bose–Hubbard Hamiltonian

The standard Bose–Hubbard evolution is simulated by an energy-preserving first-order Lie–Trotter circuit on a photon-number cutoff. The guarantee also extends to states with bounded expected photon number, with polynomial complexity under the stated fixed-parameter conditions.

F.1 Standard Bose–Hubbard Hamiltonian

  • For fixed ν, J, U, and µ, the number of Trotter steps is polynomial in m, t, K, and 1/ε.The theorem supplies an energy-preserving circuit approximating the Bose–Hubbard evolution on the cutoff subspace.Source: §F.1 · p. 110
  • The first-order Lie–Trotter circuit consists of allowed energy-preserving gates generated by hopping terms and on-site polynomials.The cutoff is the subspace with at most K total photons.Source: §F.1 · pp. 110–111
  • The simulation extends to states with bounded expected photon number, with complexity poly(m, t, Ē, 1/ε).The corollary obtains this guarantee by choosing a photon cutoff based on the mean-energy bound and target error.Source: §F.1 · p. 112

F.2 Active Bose–Hubbard Hamiltonian

A coherent-state ancilla lets the active Bose–Hubbard dynamics be simulated using energy-preserving Hamiltonians. The total error combines the ancilla approximation with product-formula error on fixed total-photon sectors.

F.2 Active Bose–Hubbard Hamiltonian

  • The active Bose–Hubbard simulation uses one coherent-state ancilla and an energy-preserving Hamiltonian on the system plus ancilla.Displacing the ancilla converts the driven dynamics into the target Hamiltonian plus a remainder.Source: §F.2 · pp. 116–117
  • The theorem treats normalized system states with finite expected photon number and nonnegative evolution time.The initial system–ancilla state has mean total photon number equal to the system mean plus α^2.Source: §F.2 · pp. 117, 119
  • Drive terms sharing the ancilla need not commute even when they act on different system modes, so the analysis bounds commutators over all term pairs.This is more conservative than exploiting only nonzero commutators.Source: §F.2 · p. 118
  • The total error has an ancilla approximation term scaling as O(α^-1) and a product-formula term scaling as O(t^2/r) on each fixed total-number sector.The Trotter factors are energy-preserving gates.Source: §F.2 · pp. 118–119

F.3 Comparison with related approaches

The analysis complements the truncation framework of Tong et al. [TAM+22] by using exact total-number conservation to simulate standard Bose–Hubbard dynamics directly in an invariant cutoff space. Compared with prior gate decompositions, it preserves particle number in each factor and also treats a driven extension using a coherent ancillary mode.

F.3 Comparison with related approaches

  • Tong et al.'s truncation bounds [TAM+22] require bounded or well-approximated initial local occupation and local-growth conditions on the number-changing Hamiltonian part.Under those hypotheses, their local cutoff can scale polylogarithmically with 1/ε; their direct theorem does not cover arbitrary Bose–Hubbard dynamics.Source: §F.3 · p. 119
  • The first-order product-formula simulation uses polynomially many Trotter steps in m, t, K, and 1/ε, with linear dependence on 1/ε.It operates directly in the invariant cutoff space H(m)≤K by exploiting exact total-number conservation.Source: §F.3 · p. 120
  • Unlike the Bose–Hubbard gate decomposition of Kalajdzievski et al. [KWR18], every factor in this standard-Hamiltonian decomposition preserves total particle number.The two constructions use different gate sets and treatments of the infinite-dimensional Hilbert space, so their gate-count scalings are not directly interchangeable.Source: §F.3 · p. 120
  • For driven Bose–Hubbard dynamics, a single coherent ancillary mode replaces direct implementation of the number-changing drive with an energy-preserving interaction.Theorem F.4 quantifies the additional approximation error through its dependence on the coherent-state amplitude α.Source: §F.3 · p. 120

G Approximate Hermitian diagonalization

The section gives a self-contained Jacobi-based algorithm for approximately diagonalizing rational complex Hermitian matrices, with explicit iteration and working-precision bounds. Its stated running time is polynomial in the matrix dimension and input and precision parameters.

G Approximate Hermitian diagonalization

  • Theorem G.1 gives a deterministic approximate diagonalization algorithm for a norm-bounded rational Hermitian matrix, with running time d^3poly(b + p + log d) bit operations.The analysis is self-contained for complex Hermitian matrices and provides explicit iteration and working-precision bounds.Source: G Approximate Hermitian diagonalization · pp. 120–121
  • Each Jacobi step selects the largest-magnitude off-diagonal entry and applies a unitary rotation on its two coordinates to eliminate it.The resulting matrix is diagonal on that selected 2×2 block.Source: G Approximate Hermitian diagonalization · p. 121
  • Each Jacobi iteration reduces off(·) by a factor of ϱ, and the stated stopping conditions ensure termination after at most kmax rotations.The conditions include dϱ^kmax ≤ ε/16 and d^2η ≤ ε/100.Source: G Approximate Hermitian diagonalization · pp. 121–122
  • The implementation stores entries as dyadic rationals and rounds the input matrix and approximate rotations while preserving Hermitian symmetry.The analysis bounds rotation and product-rounding errors and establishes that fixed precision of O(p + log d) bits suffices.Source: G Approximate Hermitian diagonalization · pp. 121–122
  • The accumulated rounding and approximation errors are bounded so the final diagonalization guarantees in Eqs. (657) and (658) are each at most ε.The argument uses α ≤ ε/100 and (k* + 1)η ≤ ε/100.Source: G Approximate Hermitian diagonalization · p. 122
  • An indexed max-heap supports largest-entry selection and updates, yielding the stated d^3poly(b + p + log d) total bit complexity.Each rotation changes O(d) matrix entries, and the heap and related quantities can be updated in O(d log d) scalar operations per iteration.Source: G Approximate Hermitian diagonalization · pp. 120, 123

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