Stochastic Quantum Hamiltonian Descent

Hongyi Zhou; Shengminjie Chen; Sirui Peng; Xiaoming Sun

REVIEW 3 major objections 4 minor 46 references

Stochastic Quantum Hamiltonian Descent

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Stochastic quantum descent: provable convergence at one query per step

desk verdict Genuinely new stochastic quantum optimization idea, but the convergence proof and the discrete-to-continuous bridge each have a load-bearing sign/extension error; worth refereeing after the authors fix them. read the letter →

arxiv 2507.15424 v1 pith:LUIQQG4X submitted 2025-07-21 quant-ph

Sirui Peng , Shengminjie Chen , Xiaoming Sun , Hongyi Zhou This is my paper

classification quant-ph MSC 81P6890C2568Q1281S22 PACS 03.67.Ac03.65.Yz

keywords stochasticgradientdescentquantumHamiltonianLindbladmasterequationconvexoptimizationweakapproximationgate-basedalgorithmfinite-sumtunneling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to give gradient-based quantum optimization the same practical advantage that stochastic gradient descent gives classical machine learning: instead of querying the full objective $f = (1/m)\sum_{j=1}^m f_j$ at every step, the algorithm queries one randomly chosen component $f_j$. The authors propose a continuous-time open quantum system, a Lindblad master equation, as the quantum analogue of continuous-time SGD, and prove that its expected-loss error relative to the global minimum decays as $O(e^{-\beta(\tau)} + \eta \sigma_f^*)$ for convex, smooth components, where $\sigma_f^* = E_j[\|\nabla f_j(x^*) - \nabla f(x^*)\|^2]$ is the gradient noise at the optimum. They then give a gate-based discrete algorithm, Stochastic Quantum Hamiltonian Descent (SQHD), and prove it is an order-2 quantum weak approximation of that continuous dynamics, so the same convergence guarantee carries over to a circuit that can be run on near-term devices. The significance is that SQHD plausibly combines the global-exploration ability of quantum Hamiltonian descent with the per-iteration cost of classical SGD, a regime that neither method reaches alone.

What carries the argument

The load-bearing object is the Lindblad generator $L_{LS}(t) = u(t)L_{GD}(t) + u(t)^2\eta L_{NOISE}(t)$, where $L_{GD}$ is the Hamiltonian evolution of QHD with $H(t) = e^{\psi(t)}(-\tfrac{1}{2}\Delta) + e^{\chi(t)}\hat{f}$, and $L_{NOISE}$ is the dissipation $\frac{1}{2}([\hat{f},[\hat{f},\cdot]] - \frac{1}{m}\sum_j [\hat{f}_j,[\hat{f}_j,\cdot]])$ measuring the excess diffusion of stochastic over full-batch evolution. The proof of Theorem 1 uses the QHD Lyapunov function $E(\tau) = \langle \tfrac{1}{2}(\hat{x} + e^{-\gamma(\tau)}\hat{p})^2 + e^{\beta(\tau)}\hat{f}\rangle_\tau$ with the strong ideal scaling conditions $\psi = \alpha - \gamma$, $\chi = \alpha + \beta + \gamma$, $\dot{\beta} = \dot{\gamma} = e^{\alpha}$, and Lemma 5 bounding the expected squared gradient deviation by $8dL_{\max}^2 + \sigma_f^*$. The proof of Theorem 2 rests on a quantum generalization of the classical weak-approximation framework: the continuous and discrete processes are compared in trace distance, with the error controlled by smoothness of the evolved state up to order 6, the vector-norm scaling for unbounded Hamiltonians, and a periodic-extension lemma that moves smooth functions on the box to the torus.

What would settle it

On the one-dimensional convex objective $f(x) = x^3 - 3x$ restricted to $[-1,1]$ (convex on that interval, with a second-derivative mismatch $f''(-1) \neq f''(1)$), compute the trace distance $\|\Lambda_{LS}(0,N\eta)[\rho_0] - \Lambda_{dSQHD}(N,\eta)[\rho_0]\|_1$ for $\eta = 0.01, 0.005, 0.0025$ at fixed total time $T=N\eta$. If the trace distance does not scale as $O(\eta^2)$ — for instance, if it grows with $T$ or the scaling exponent drops below 2 — then the periodic-extension or smoothness-propagation assumption behind Theorem 2 fails, and the discrete algorithm does not inherit the continuous convergence bound.

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Extended reading notes

Core claim

On convex and smooth finite-sum problems, the dynamics called Stochastic Quantum Hybrid Dynamics drive the expected objective value to the global minimum at rate $O(e^{-\beta(\tau)} + \eta \sigma_f^*)$, where $\tau$ is the effective time under the learning-rate schedule and $\sigma_f^* = E_j[\|\nabla f_j(x^*) - \nabla f(x^*)\|^2]$ is the gradient noise at the minimizer. The dissipation in the Lindblad equation is engineered to reproduce exactly the second-order difference between applying the full phase operator $e^{-i\eta \hat{f}}$ and the averaged stochastic phases $(1/m)\sum_j e^{-i\eta \hat{f}_j}$, so the noise term $\frac{1}{2}([\hat{f},[\hat{f},\rho]] - \frac{1}{m}\sum_j [\hat{f}_j,[\hat{f}_j,\rho]])$ captures the diffusion introduced by stochastic sampling. The discrete algorithm applies, in each epoch, the Trotter-style sequence of kinetic and stochastic potential unitaries $\exp(-i\frac{\eta}{2} a_j(-\Delta/2))\exp(-i\eta b_j \hat{f}_{\xi_j})\exp(-i\frac{\eta}{2} a_j(-\Delta/2))$; Theorem 2 proves this channel is an order-2 weak approximation of the continuous dynamics under smoothness assumptions on the evolved state and the components, so the gate-based SQHD inherits the convergence bound with an additional $O(\eta^2)$ discretization error.

Load-bearing premise

The argument assumes that the quantum state remains smooth (all derivatives up to order 6 stay bounded) for the entire run, and that every smooth component function on the box $[-1,1]^d$ can be extended periodically with all boundary derivatives matching; a component with mismatched boundary derivatives, or a state whose derivatives blow up over time, would break the $O(\eta^2)$ approximation and the consequent convergence claim.

Editorial extensions

If this is right

SQHD solves a convex finite-sum problem with expected loss gap $O(e^{-\beta(\tau)} + \eta \sigma_f^*)$, giving an exponential-in-effective-time phase followed by a classical noise floor linear in the learning rate; for SGDM-style schedules this reads $O(1/t + \eta \sigma_f^*)$ with the crossover at $t^* \sim \eta^{-1}$.
The per-iteration cost drops from a full-batch query of $f$ to a single coherent query of one component $f_j$, so the method inherits SGD's scalability to large $m$ while retaining QHD's tunneling-based exploration.
Because the discrete channel is an order-2 weak approximation of the continuous Lindblad dynamics, the convergence guarantee is robust to discretization: the $O(\eta^2)$ approximation error is additive and does not break the $O(e^{-\beta(\tau)} + \eta \sigma_f^*)$ bound.
The algorithm avoids direct Lindbladian simulation, which would require querying the full objective and simulating complex dissipative terms; it is built from standard real-space simulation primitives, making it amenable to near-term hardware.

Reading between the lines

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

A testable extension suggested by the proof: the $O(\eta^2)$ weak approximation should degrade to a lower-order scaling if any component $f_j$ lacks matching derivatives at the box boundary, since the periodic-extension lemma then fails; this could be checked numerically on $f(x)=x^3-3x$ on $[-1,1]$.
The derivation isolates a general recipe: quantize any stochastic gradient method by matching first-order evolution and casting the variance of component phases as Lindblad dissipation. Applied to adaptive methods such as Adam or RMSProp, this could yield \u201cStochastic Quantum Hamiltonian\u201d variants with different noise geometries.
The phase transition at $t^* \sim 1/\eta$ between fast quantum descent and noisy classical fluctuation suggests an operating principle for the method: run SQHD slightly past $t^*$ and then switch to a classical local method, rather than letting the quantum noise floor dominate. The paper does not discuss this hybrid handoff.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a real and plausible new algorithm, but the proof chain currently has two load-bearing gaps, so the result should not be accepted as-is.

What is genuinely new: the Lindblad generator with jump operators proportional to the component functions f_j, and the gate-based SQHD algorithm that approximates the open-system dynamics without simulating the Lindbladian directly. That is a natural quantum analogue of continuous-time SGD, and the weak-approximation framework is the right tool for connecting the Trotter-like discrete update to the continuous limit. If the convergence theorem holds, SQHD is a useful stochastic counterpart to QHD with one component query per iteration. The authors are also honest about the trade-off between per-iteration cost and slower convergence; that is welcome.

Now the soft spots, in proportion to how serious they are. First, the sign of the noise term is wrong in a way that breaks the central proof. In Appendix B, with A_j = u√η e^χ f_j and γ_jk = δ_jk/m − 1/m^2, the Lindblad dissipator evaluates to minus the L_NOISE defined in Eq. (1). The proof of Theorem 1 then uses the positive sign to obtain the desired inequality; direct computation gives the opposite. The reader flagged Eq. (C42) and the commutator identity (C50)–(C52), but the problem is deeper: the generator as written is not the Lindblad generator they claim. This is not cosmetic; the Lyapunov argument rests on it.

Second, the bridge between the continuous dynamics and the discrete algorithm is broken by the periodic-extension assumption. Lemma 7 requires a periodic function on R^d, but a generic smooth function on the box C = [-1,1]^d has no smooth period-2 extension unless derivatives match on opposite faces, and the grid in Appendix E uses exactly that period. Extending with a larger period changes the domain the discrete algorithm simulates; worse, a nonconstant convex function cannot be extended periodically while staying convex on R^d, which Theorem 1's proof needs. So Theorem 2, as stated, does not follow, and the O(η^2) weak-approximation result is unsupported.

Third, the numerics are thin: ten samples, no error bars, no code. That is minor compared with the proof gaps, but it means the nonconvex advantage is not established.

Who should read this? Researchers working on quantum optimization and quantum machine learning. The core construction is clever and worth knowing even if the proofs need repair. It deserves a serious referee, not a desk reject, but I would not cite it in its current form. Send it to referees and require the sign errors and the periodic-extension assumption to be fixed before publication.

Referee Report

3 major / 4 minor

Summary. The paper introduces Stochastic Quantum Hamiltonian Descent (SQHD), a gate-based quantum algorithm intended to approximate a continuous Lindbladian dynamics that is described as a quantum analogue of continuous-time SGD. The central theoretical claims are Theorem 1, which asserts that the continuous dynamics converge for convex and smooth cost components with expected-loss bound O(e^{-β(τ)} + η σ*_f), and Theorem 2, which asserts that the discrete gate-based channel is an order-2 quantum weak approximation of the continuous dynamics. Together these are meant to imply that the implementable algorithm inherits the convergence bound while querying only one component function per iteration. The paper also reports numerical experiments on several non-convex benchmarks comparing SQHD with QHD and SGDM.

Significance. If the proof chain were sound, the paper would give a natural stochastic-gradient counterpart of Quantum Hamiltonian Descent with per-iteration cost of one component query, which is a notable and timely contribution. The use of the weak-approximation framework from stochastic modified equations is appropriate, and the paper is largely self-contained, with no fitted parameters in the convergence analysis and a broad set of numerical experiments. However, the central proof has two load-bearing gaps: an algebraic sign error in the Lyapunov computation of Theorem 1, and a domain/periodicity incompatibility in the proof of Theorem 2 that prevents the discrete algorithm from inheriting the continuous convergence result for the stated class of box-domain convex problems. Because the latter gap concerns the central claim of the paper and cannot be repaired by a local correction, the manuscript in its current form does not establish its advertised convergence guarantee.

major comments (3)

[Appendix C, Eqs. (C50)–(C52)] The nested commutator [g,[Δ,g]] is computed incorrectly. A direct calculation gives [g,[Δ,g]] = -2||∇g||², and equivalently [g,[Δ,g]] = [g, Δg + 2(∇g)·∇] without the minus sign written in Eq. (C50). The sign of the noise term in Eq. (C42) is consistent with the Lindblad generator in Eq. (1) and Appendix B, but the algebra in Eqs. (C50)–(C52) implies [g,[E,g]] = -e^{-2γ}||∇g||² for the Lyapunov function E, making the noise contribution in Eq. (C44) negative rather than positive. The final sign in Eq. (C44) is the one needed for Theorem 1 and agrees with the correct commutator value, so the theorem may be salvageable, but the proof as written is internally inconsistent.
[Appendix D, Lemma 7 and following paragraph] The assertion that any smooth f:C→R can be extended periodically while preserving smoothness is false for the period-2 torus actually used by the algorithm. For f(x)=x² on C=[-1,1], one has f'(-1)=-2 and f'(1)=2, so no C¹ period-2 extension exists. Extending with a larger period would place the continuous dynamics on a different torus, which the period-2 DFT Laplacian in Eqs. (E5)–(E8) does not simulate. Since the proof of Theorem 2 uses Lemma 7 to conclude that the discrete states ρ_k remain smooth up to order 6, the error recursion in Eqs. (D42)–(D45) and the consequent O(η²) weak-approximation bound are not justified.
[Theorems 1 and 2, Lemmas 5 and 7] The hypotheses of Theorem 1 and the hypotheses needed for Lemma 7 are incompatible for generic nonconstant convex components. Theorem 1 and Lemma 5 use convexity of the components (Eq. (C55) and Lemma 5), while the proof of Theorem 2 requires, through Lemma 7, that each f_j be extendable to a smooth periodic function on R^d. A nonconstant convex function on R^d cannot be periodic: a periodic convex function is constant. Therefore no nonconstant convex component satisfies both requirements, and the combination of Theorems 1 and 2 does not establish convergence of the gate-based SQHD for the stated class of convex finite-sum problems on the box.

minor comments (4)

[Section III.B, Lemma 2 proof] The first equality in the proof, |tr(O(˜ρ−ρ))| = ||O(˜ρ−ρ)||_1, is false; the correct statement is |tr(OA)| ≤ ||O A||_1 ≤ ||O|| ||A||_1. The argument goes through after replacing the equality by an inequality.
[Appendix C, Lemma 5] The inequalities in Lemma 5 are stated for all x∈R^d, although the functions f_j are defined only on C=[-1,1]^d. The proof should either restrict the domain to C or explicitly justify the extension used.
[Appendix C, Eqs. (C56)–(C62)] The constant G appearing in Eqs. (C57) and (C62) is not defined in the statement of Theorem 1, and the numerical factor multiplying dLmax changes from 4 in Eq. (C57) to 1 in Eq. (C62) without explanation.
[Appendix D, Lemma 6 proof] The equality ||Σ_j p_j A_j||_1 = Σ_j p_j ||A_j||_1 in the proof of Lemma 6 is not true in general; the inequality ≤ follows from the triangle inequality and is sufficient for the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the SQHD convergence and approximation results are derived from external Lyapunov and weak-approximation frameworks; the only self-citation is non-load-bearing background.

full rationale

The paper's load-bearing results do not reduce to their inputs by construction. Theorem 1 extends the QHD Lyapunov analysis of [5] (Leng, Hickman, Li, Wu; no author overlap) to the Lindblad generator (1); the O(eta sigma*_f) term arises from the smoothness/convexity bound in Lemma 5 (Eq. C22/C35), where sigma*_f is the problem's gradient noise parameter, not a fitted quantity renamed as a prediction. Theorem 2 explicitly identifies itself as 'a quantization of [29, Theorem 1]' and derives the O(eta^2) trace-distance bound by matching Taylor expansions of the continuous and discrete channels, with remainder control from Lemma 8 and external smoothness lemmas [32,33]; no equation in the discrete proof is set equal to the convergence bound by definition. The only author self-citation, [25] (Peng, Sun, Zhao, Zhou), appears in the background list '[22-25]' for direct Lindbladian simulation and is explicitly circumvented by the SQHD algorithm, so it is not load-bearing. The periodic-extension assertion used with Lemma 7 is a mathematical-support concern (whether a box function admits a period-2 smooth periodic extension preserving convexity), not a circularity, because the target convergence bound is not assumed, fit, or otherwise encoded in the hypotheses. The Discussion's caveats (cost/convergence trade-off, hyperparameter sensitivity) are stated limitations rather than circular moves.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on convexity and smoothness assumptions, a smoothness-preservation lemma whose periodic-extension requirement is nontrivial for box domains, and a strong scaling condition on schedules. No new physical entities are introduced, and no parameters are fitted to data; the only hand-chosen quantities are standard hyperparameters such as learning rate and Hamiltonian schedules.

free parameters (2)

learning rate eta = 0.01, 0.005, 0.0025 (numerical sweeps)
Chosen by hand as a hyperparameter; the convergence bound scales linearly with eta and with gradient noise sigma*_f.
Hamiltonian coefficient schedules e^psi, e^chi, u(t) = NAGD: e^psi=2t^-3, e^chi=2t^3, u=1; SGDM: e^psi=t^-2, e^chi=2t, u=1/2
Hand-chosen to match classical NAGD and SGDM schedules; not fitted to data. The strong ideal scaling condition constrains these schedules in Theorem 1.

assumptions (5)

domain assumption Each f_j is convex and smooth up to order 2 with bound L_j on C=[-1,1]^d
Theorem 1 requires convexity and smoothness of every component; Lemma 5 bounds gradient noise using these properties.
domain assumption Initial state and evolved states remain smooth up to order 6 on [0, N*eta]
Theorem 2's O(eta^2) approximation and Lemma 9's bounded nested commutators rely on this; it is not proven for general initial states or long times.
ad hoc to paper f on C can be extended to a periodic function on R^d preserving smoothness
Lemma 7 needs periodicity to apply [33, Lemma 6.2]; arbitrary smooth functions on a box are not periodic with matching derivatives, so this is a nontrivial assumption.
domain assumption Strong ideal scaling condition psi=alpha-gamma, chi=alpha+beta+gamma, beta_dot=gamma_dot=e^alpha
Needed for the Lyapunov analysis of Theorem 1; restricts admissible coefficient schedules.
standard math Lindblad master equation validity and standard trace-norm inequalities
Used to show Eq.(1) is a quantum channel and to control approximation errors in Lemma 2 and Theorem 2.

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Cite this review

Pith. "Pith review of Stochastic Quantum Hamiltonian Descent." pith.science (2026). https://pith.science/paper/LUIQQG4X

@misc{pith2026250715424,
  author       = {Pith},
  title        = {Pith review of: Stochastic Quantum Hamiltonian Descent},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LUIQQG4X}},
  note         = {Machine review of arXiv:2507.15424}
}

read the original abstract

Stochastic Gradient Descent (SGD) and its variants underpin modern machine learning by enabling efficient optimization of large-scale models. However, their local search nature limits exploration in complex landscapes. In this paper, we introduce Stochastic Quantum Hamiltonian Descent (SQHD), a quantum optimization algorithm that integrates the computational efficiency of stochastic gradient methods with the global exploration power of quantum dynamics. We propose a Lindbladian dynamics as the quantum analogue of continuous-time SGD. We further propose a discrete-time gate-based algorithm that approximates these dynamics while avoiding direct Lindbladian simulation, enabling practical implementation on near-term quantum devices. We rigorously prove the convergence of SQHD for convex and smooth objectives. Numerical experiments demonstrate that SQHD also exhibits advantages in non-convex optimization. All these results highlight its potential for quantum-enhanced machine learning.

Figures

Figures reproduced from arXiv: 2507.15424 by the authors.

**Figure 2.** FIG. 2: Comparison of SQHD, QHD, and SGDM on objective functions [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3: Conceptual schema illustrating the relationship [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: Results of Stochastic Quantum Hamiltonian Descent and Stochastic Quantum Hybrid Dynamics on all test [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

**Figure 5.** Figure 5: FIG. 5: Results of SQHD, QHD, and SGDM on different objective functions varying with learning rate [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

**Figure 6.** Figure 6: FIG. 6: Results of SQHD, QHD, and SGDM on different objective functions varying with resolution [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

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This represents the ideal, noiseless component of the dynamics, LGD(t)[σ] = −i[H(t), σ]

A Gradient Descent Term ( LGD): This term drives the system along the desired optimiza- tion path according to the standard Schroedinger equation, governed by the Hamiltonian H(t) = eψ(t) − 1 2 ∆ + eχ(t) ˆf , where ∆ = ∂2 ∂x2 1 + ∂2 ∂x2 2 + · · ·+ ∂2 ∂x2 d is the Laplace operator. This represents the ideal, noiseless component of the dynamics, LGD(t)[σ] =...

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I + η LGD + η 2 ˙LGD + η2 LN OISE+ 1 2 L2 GD # [ρk] + O(η3) (D40) =

I. Goodfellow, Y. Bengio, and A. Courville, Deep Learn- ing (The MIT Press, 2016). 9 Appendix A: Problem Setting and Assumptions We consider the d-dimensional unconstrained finite-sum optimization problem on the domain C = [−1, 1]d ⊂ Rd, expressed as min x∈C f (x), f(x) = 1 m ...

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The computational basis in the Hilbert space is labeled by ˜C = [x1,

Implementation The numerical simulation is conducted on a space-discretized Hilbert space with d log2 nr qubits, where nr is the resolution of the grid. The computational basis in the Hilbert space is labeled by ˜C = [x1, . . . , xd], xj ∈ −1 + 2k + 1 nr , k= 0, 1, . . . , nr ...
[42]

Setting a. Test functions The first function we consider is the rotated high-dimensional double-well function: FU,s(⃗ x) = F 1 s U ⃗ x , F(⃗ x) = 1 d dX j=1 w(xj), (E15) where U is a d-dimensional orthogonal matrix and w : R → R is a smooth non-convex function with 2 local min...
[43]

V alidation of the approximation result We compare the result of the SQHD algorithm with direct simulation of Stochastic Quantum Hybrid Dynamics (1) in Figure 4 to validate the approximation result (Theorem 2). FIG. 4: Results of Stochastic Quantum Hamiltonian Descent and Stoc...
[44]

The results in Figure 5 show that SQHD with a smaller learning rate has smaller fluctuation

Rule of learning rate We run SQHD, QHD, and SGDM with the default setting on all test functions with resolution nr = 32 except that the iteration number is set to N = 8000, 16000, 32000. The results in Figure 5 show that SQHD with a smaller learning rate has smaller fluctuatio...
[45]

The results in Figure 6 show that the solution quality of SQHD is not affected by the change of resolution

Rule of resolution We run QHD, SQHD, and SGDM with default settings except that the resolution is set to nr = 32, 128. The results in Figure 6 show that the solution quality of SQHD is not affected by the change of resolution. Nevertheless, SQHD shows a slower converging proce...
[46]

General comparison We compare the results of QHD, SQHD, and SGDM with default settings except for resolution nr = 128 and iteration number N = 32000, and the results are shown in Figure 2. The results show that, overall, SQHD achieves comparable solution quality to QHD while i...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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This represents the ideal, noiseless component of the dynamics, LGD(t)[σ] = −i[H(t), σ]

A Gradient Descent Term ( LGD): This term drives the system along the desired optimiza- tion path according to the standard Schroedinger equation, governed by the Hamiltonian H(t) = eψ(t) − 1 2 ∆ + eχ(t) ˆf , where ∆ = ∂2 ∂x2 1 + ∂2 ∂x2 2 + · · ·+ ∂2 ∂x2 d is the Laplace operator. This represents the ideal, noiseless component of the dynamics, LGD(t)[σ] =...

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1: Overview of the SQHD method

A Stochastic Noise Term ( LN OISE): This term explicitly models the diffusive effects intro- duced by the stochastic updates, as derived in our 3 FIG. 1: Overview of the SQHD method. (Top) The SQHD method is designed for unconstrained finite-sum optimization problems, where queries to individual objective functions f1, . . . , fm (rather than the full obj...

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Prepare the initial guess state ρ0

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, N− 1: (a) Calculate the discrete parameters aj = exp(ψ((j + 1/2)η)), bj = exp(χ((j + 1/2)η))

For epoch j = 0, 1, . . . , N− 1: (a) Calculate the discrete parameters aj = exp(ψ((j + 1/2)η)), bj = exp(χ((j + 1/2)η)). (b) Update with unitary exp( −i η 2 aj(−∆/2)). (c) Update with unitary exp( −iηbj ˆfξj ), where ξj is independently and uniformly drawn from {1, . . . , m}. (d) Update with unitary exp( −i η 2 aj(−∆/2))

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This algorithm outputs a value x such that E[f (x)] is approximately D ˆf E t where t = N η

Measure the final state with the position operator ˆx =R x x |x⟩ ⟨x| dx and output the measured value. This algorithm outputs a value x such that E[f (x)] is approximately D ˆf E t where t = N η. The update UdSQHD (N, η; ξ) (Step 2 of the algorithm) is similar to a second-order Trotter-Suzuki decomposition of the Hamil- tonian dynamic in the Lindblad equa...

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[6] [6]

We believe a similar phase exists for SQHD; however, excessive ran- domness can weaken its effectiveness

suggest that its effectiveness, particularly on non- convex problems, stems from a ”global search” phase inherent in the optimization dynamics. We believe a similar phase exists for SQHD; however, excessive ran- domness can weaken its effectiveness. This stochasticity, which is governed by hyperparameters such as the learn- ing rate and the Hamiltonian co...

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H. E. Robbins, A stochastic approximation method, An- nals of Mathematical Statistics 22, 400 (1951)

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Wibisono, A

A. Wibisono, A. C. Wilson, and M. I. Jordan, A vari- ational perspective on accelerated methods in optimiza- tion, Proceedings of the National Academy of Sciences 113, E7351 (2016)

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