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arXiv:2607.16996

Encoding Choices and Fault-Tolerant Resource Estimates for Digital Quantum Hamiltonian Descent

Chenxu Liu, Meng Wang, Mingze Li, Muqing Zheng, Samuel Stein, Yousu Chen

quant-phSubmitted Jul 18, 2026• Announced Jul 21, 2026• 0 citations
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Abstract

Quantum Hamiltonian descent (QHD) formulates continuous optimization as time-dependent quantum dynamics, where a kinetic term drives exploration and a potential term encodes the objective function. Digital implementations of QHD require encoding the search space into qubits, and this choice can shift the dominant cost among logical qubits, circuit depth, non-Clifford rotations, and potential synthesis. In this work, we present an encoding-aware resource analysis comparing one-hot and binary amplitude encodings for QHD. We derive gate-count scalings, construct and validate circuits against classical \Sch-equation solvers, and estimate Clifford+$R_z$ and fault-tolerant Clifford+$T$ resources on benchmark optimization problems. Binary encoding reduces the data register from $O(dN)$ to $O(d\log N)$ qubits and gives comparable asymptotic scaling for both kinetic and potential evolutions. Across all benchmark problems studied, binary encoding also uses fewer $R_z$ rotations than one-hot encoding, making it the preferred option for fault-tolerant implementations where arbitrary rotations dominate the cost. Kinetic approximations based on low-momentum spectra and approximate QFTs can further reduce the binary kinetic cost to polylogarithmic scaling. However, for targets such as Ackley, potential synthesis can dominate the total cost and reduce the benefit of kinetic approximations. These results suggest that exploiting the analytic structure of the target function to compile the potential evolution in QHD more efficiently is needed for further resource reductions.

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28 pages, 14 figures, 3 tables

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Quantum Hamiltonian descent (QHD) formulates continuous optimization as time-dependent quantum dynamics, where a kinetic term drives exploration and a potential term encodes the objective function. Digital implementations of QHD require encoding the search space into qubits, and this choice can shift the dominant cost among logical qubits, circuit depth, non-Clifford rotations, and potential synthesis. In this work, we present an encoding-aware resource analysis comparing one-hot and binary amplitude encodings for QHD. We derive gate-count scalings, construct and validate circuits against classical \Sch-equation solvers, and estimate Clifford+RzR_zRz​ and fault-tolerant Clifford+TTT resources on benchmark optimization problems. Binary encoding reduces the data register from O(dN)O(dN)O(dN) to O(dlog⁡N)O(d\log N)O(dlogN) qubits and gives comparable asymptotic scaling for both kinetic and potential evolutions. Across all benchmark problems studied, binary encoding also uses fewer RzR_zRz​ rotations than one-hot encoding, making it the preferred option for fault-tolerant implementations where arbitrary rotations dominate the cost. Kinetic approximations based on low-momentum spectra and approximate QFTs can further reduce the binary kinetic cost to polylogarithmic scaling. However, for targets such as Ackley, potential synthesis can dominate the total cost and reduce the benefit of kinetic approximations. These results suggest that exploiting the analytic structure of the target function to compile the potential evolution in QHD more efficiently is needed for further resource reductions.
Quantum Hamiltonian Descent (QHD) is a continuous optimization algorithm based on simulating a time-dependent quantum Hamiltonian whose potential energy encodes the objective function and whose kinetic energy promotes exploration through quantum interference and tunneling. While QHD is formulated for unconstrained optimization, many real-world optimization problems are constrained and highly nonconvex. In this paper, we benchmark AL-QHD, a hybrid framework that embeds QHD within the Augmented Lagrangian Method (ALM), thereby solving a sequence of unconstrained subproblems while using ALM to enforce constraints. We evaluate AL-QHD on standard nonconvex test functions and use iterative refinement to improve solution accuracy at fixed per-run qubit cost. We also perform a gate-based resource analysis on ACOPF-derived power system subproblems constructed from power-network data to estimate the quantum-computer scale required for practical applications. Resource estimates on Texas7k-derived ACOPF instances show steep hard-gate scaling, reaching ∼4.46×107\sim 4.46 \times 10^7∼4.46×107 entangling gates in a NISQ-oriented model and ∼9.42×108\sim 9.42 \times 10^8∼9.42×108 T gates in a fault-tolerant model at ∼5.3×102\sim 5.3 \times 10^2∼5.3×102 active variables. These results suggest that AL-QHD is a useful framework for studying constrained nonconvex optimization with QHD, but that practical ACOPF-scale applications would likely require large-scale fault-tolerant quantum hardware.
Parametric quantum circuits are usually trained by classical, local optimizers, which use the quantum computer only to evaluate the cost function, or its gradient, one point at a time. Quantum Hamiltonian Descent (QHD) takes a different route: in this quantum algorithm for continuous optimization, the parameters become the position of a quantum particle, with the cost function as its potential, and two time schedules control kinetic and potential energy so that the wavefunction explores the whole landscape and tunnels towards the global minimum. QHD, however, assumes a classically computable cost function, and its digital implementation is poorly understood: standard error bounds grow exponentially with the number of qubits per parameter, and the schedules are borrowed from classical optimization. Here we propose to train parametric circuits with QHD entirely on the quantum computer, through an online step in which the qubits encoding the parameters coherently control the circuit on a second register. We prove that, for cost functions with a smooth periodic extension, including those of parametric circuits, the simulation error admits a bound independent of the grid resolution, and that for circuits the bound grows at most linearly with the number of parameters. We also show that the ideal dynamics depends on the two schedules only through a single curve, which controls the adiabaticity of the search, and that the remaining freedom can be used to minimize the gate count. Simulations on benchmark functions and parametric circuits show that, at equal cost, schedules designed in this way markedly increase the probability of finding the global minimum.
Finite local Hilbert-space truncations arise naturally in quantum simulations of lattice field theories and motivate qudit encodings, but their fault-tolerant advantage over qubit encodings remains unclear. We compare the non-Clifford cost of implementing quadratic diagonal evolutions, exemplified by U=e−itφx2U=e^{-itφ_x^2}U=e−itφx2​ in a uniform field-amplitude discretization of a real scalar field, using either one logical ddd-level qudit or nb=⌈log⁡2d⌉n_b=\lceil \log_2 d\rceilnb​=⌈log2​d⌉ logical qubits. We analyze two standard settings: product-formula simulation and LCU/block encoding, taking the resource metric to be the number of non-Clifford gates after synthesis into a discrete logical gate set. Because tight synthesis bounds for general single-qudit rotations are not known, we express the qudit constructions in terms of embedded two-level SU(2)SU(2)SU(2) rotations and derive explicit finite-ddd break-even conditions for their synthesis cost; these serve as compiler targets for when qudit encodings can outperform the qubit baseline. Within the constructive models studied here, product-formula implementations would require an exponentially stronger per-primitive synthesis advantage for qudits to win asymptotically, while in the LCU setting the qubit encoding is asymptotically cheaper in ddd. Nevertheless, the finite-ddd threshold analysis identifies low dimensional regions in which qudits can yield meaningful constant-factor savings, particularly for LCU-based implementations. As a secondary analysis of the LCU construction, we use an idealized negligible-overhead qubit-qudit code-switching model to give an absolute TTT-count comparison, and reinterpret the savings as an allowable per-switch overhead budget.
Fault-tolerant quantum computing typically requires the transpilation of arbitrary quantum circuits into a finite, universal gate set, such as Clifford+T. As a baseline, Diagonal approximation can be used for synthesizing single-qubit Pauli rotations, yielding an approximating sequence with TTT-count that equals 3log⁡2(1/ε)3 \log_2(1/ε)3log2​(1/ε) for a target precision εεε. Magnitude Approximation can reduce the TTT-count to only 1log⁡2(1/ε)1 \log_2(1/ε)1log2​(1/ε) by allowing large residual errors, which are rotations about orthogonal axes. Within a complete quantum circuit, these residual errors can then be absorbed into neighboring gates before they are approximated themselves. Determining the optimal allocation of approximation strategies within a large, multi-qubit circuit presents a significant combinatorial challenge. In this work, we present a linear-time algorithm that guarantees an optimal solution to this problem. We demonstrate that the issue of delegating Magnitude versus Diagonal approximation across a circuit maps formally to a classical 1D Ising model with a spatially varying field. By minimizing the energy of this Hamiltonian, we identify the optimal approximation configuration for each rotation without exponential overhead. Benchmarking our method against standard diagonal approximation on random quantum circuits, we observe an average reduction of 26\% in the total approximating circuit gate count, offering a significant efficiency gain for the implementation of quantum algorithms on near-term and fault-tolerant architectures.
Preparing arbitrary logical states is a central primitive for universal fault-tolerant quantum computation and the cost of encoded-state preparation contributes directly to the overall resource overhead. This makes the synthesis of efficient general-state encoding circuits an important problem, particularly with respect to two-qubit gate count and circuit depth. Yet the synthesis of such encoders has been studied less extensively than general Clifford circuit synthesis or the preparation of specific logical Pauli-eigenstates. In this work, we develop methods for synthesizing efficient encoders for arbitrary stabilizer codes. We formulate encoder synthesis as a search over stabilizer tableaus and introduce greedy and rollout-based algorithms that exploit the freedom among stabilizer-equivalent realizations of the same encoding isometry. For code families with a modular structure, such as generalized concatenated and holographic codes, we show how large encoders can be assembled from optimized local constituent encoders, and we use SMT-based exact synthesis to obtain optimal local circuits for small instances. We further evaluate the proposed methods on a broad set of stabilizer codes, including holographic and quantum low-density parity-check (qLDPC) codes, and compare them against recent encoder-synthesis methods and existing constructions from the literature, obtaining improvements of up to 43% in two-qubit gate count and up to 70% in depth. Our results support the optimization of encoded-state preparation in several fault-tolerant quantum-computing schemes, and all methods are openly available as part of the Munich Quantum Toolkit.
This thesis develops structure-preserving methods to control and compile quantum evolutions with provable guarantees. In quantum dynamics, a fundamental resource is coherence maintained within a designated subspace; leakage – population transfer out of that subspace – is a major source of error that demands robust methods for its suppression and control. For example, maintaining the evolution within the computational (logical) subspace in quantum computing is essential for practical quantum information processing. On the dynamics side, we study adiabatic evolution under weak perturbations through effective-Hamiltonian frames. Using Bloch wave operators and Schrieffer-Wolff constructions, we obtain block-diagonal generators and prove that their evolutions remain uniformly close to the true dynamics. Moreover, we establish time-independent (“eternal”) bounds on spectral leakage that scale as 𝑂(𝛾−1) in the weak-coupling regime, where 𝛾 is the dimensionless ratio between the strength of the strong system Hamiltonian and the weak drive. The analysis covers both finite-dimensional, time-dependent settings and a novel coarse-grained framework that accommodates continuous spectra and even unbounded Hamiltonians, enabling applications to banded tight-binding models and superconducting-qubit architectures. Optimising quantum algorithms for reduced depth or fewer non-error-corrected physical gates is also key to mitigating leakage. We therefore formulate quantum compiling as exact or approximate unitary synthesis up to a physically irrelevant global phase and solve it via mixed-integer linear programming (MILP). The formulation supports multiple objectives (depth, two-qubit gate count, non-Clifford cost) and provides theoretical guarantees of optimality. To the best of our knowledge, this is the first MILP formulation that targets circuit depth directly, rather than minimising gate counts and inferring depth retroactively. The model explores the algebraic structure of the problem in order to accelerate the solver through symmetry breaking, redundancy elimination, and other mathematically sound constraints. To scale beyond exact global optimisation, we introduce a rolling-horizon strategy that compiles circuits in overlapping windows. An open-source implementation, written in the Julia language and released as QuantumCircuitOpt.jl, provides these capabilities to facilitate further research. In both cases, similarity transformations and algebraic identities reduce complexity while focusing attention on the spectral/target content, and both yield quantitative guarantees. Collectively, these contributions confront leakage, the key threat to coherent, in-subspace computation, with the ultimate aim of getting error-resilient qubits to work.