Resource-Efficient Quantum Optimization via Higher-Order Encoding

Dieter Jaksch; Frederik Koch; Joseph Doetsch; Rick Mukherjee; Shahram Panahiyan

REVIEW 3 major objections 5 minor 80 references

Resource-Efficient Quantum Optimization via Higher-Order Encoding

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Higher-order binary encodings of optimization problems use logarithmically fewer qubits than QUBO one-hot encodings, and after compilation to one- and two-qubit gates cut CNOT counts by at least 89.6% across three problem classes.

desk verdict Real per-layer resource savings and a genuinely useful HUBO construction pipeline; the headline '89.6% CNOT reduction' rests on best-of-100 QAOA minima and per-encoding lambda tuning, so treat it as optimistic rather than established. read the letter →

arxiv 2511.17545 v2 pith:TPB2ACKN submitted 2025-11-10 quant-ph

Frederik Koch , Shahram Panahiyan , Rick Mukherjee , Joseph Doetsch , Dieter Jaksch This is my paper

classification quant-ph

keywords higher-orderunconstrainedbinaryoptimizationQUBOQAOAcombinatorialone-hotencodinggateassignmentproblemmaxk-colorablesubgraphintegerprogramming

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 shows that representing each m-valued variable of a combinatorial optimization problem by ⌈log2 m⌉ qubits (a HUBO encoding) instead of m one-hot qubits (QUBO) removes the need for one-hot penalty terms and reduces circuit size. It provides a systematic Walsh-Hadamard construction of the higher-order Hamiltonian coefficients and a Gray-code circuit that implements all terms with only single- and two-qubit gates. In QAOA benchmarks on Gate Assignment, Maximum k-Colorable Subgraph, and Integer Programming problems, the authors find qubit counts that grow logarithmically instead of linearly in the number of values, and total CNOT counts to reach fixed solution thresholds that are lower by 89.6–100% than QUBO. If this holds, HUBO is a drop-in, resource-efficient replacement for QUBO in near-term quantum optimization.

What carries the argument

The central object is the value-index bit encoding: each variable i is stored in d = ⌈log2 m⌉ qubits and the projector |v⟩⟨v|_i is expanded as a sum over products of Pauli-Z operators, producing a HUBO Hamiltonian whose interacting terms act on up to 2d qubits. Two constructions carry the argument: the Walsh-Hadamard transform that converts the classical cost coefficients c1(i,v) and c2(i,j,v,w) into the HUBO coefficients J_i,S and J_{ij,S1,S2} in O(n^2 m^6) time for quadratic objectives, and the Gray-code hypercube traversal that compiles all higher-order Z-products of a d-qubit block into a circuit with one CNOT and one RZ per term — the minimal count, replacing the naive 2^d(d−2)+2 CNOTs

What would settle it

Recompute the minimum total CNOTs needed to reach the target approximation ratio using the median (not the best) of the 100 QAOA runs for both encodings, with identical penalty multipliers and identical parameter-optimization budgets; if the median saving for any tested instance falls below 89.6%, the headline savings are an artifact of selection rather than of the encoding.

Watch

Extended reading notes

Core claim

For any COP with n variables each taking one of m values, the authors construct a HUBO Hamiltonian whose ground state encodes the optimal solution, using d = ⌈log2 m⌉ qubits per variable. The bitstrings of the d qubits are read as value indices, so the one-hot penalty needed in QUBO disappears. Coefficients of the Pauli-Z expansion are obtained from the classical cost coefficients by a Walsh-Hadamard transform, with polynomial numerical complexity. The cost unitary for QAOA is built from the commuting diagonal terms using a Gray-code parity scheme, so a term acting on t qubits costs one CNOT and one RZ gate after the first, and all terms of a 2d-qubit block are implemented with 2^(2d) − 2 CN

Load-bearing premise

The 89.6–100% total-CNOT savings to reach solution thresholds come from picking the best of 100 independent QAOA runs for each encoding, after tuning penalty multipliers separately for each encoding; if that best-of-100 selection or the penalty tuning favors HUBO, the saving is larger than a fair per-instance comparison would give.

Editorial extensions

If this is right

Exponential qubit reduction: n·m qubits become n·⌈log2 m⌉, so problems with many values per variable (e.g., gate assignment with many gates) become representable.
Deterministic per-layer gate savings: compiled per-layer CNOT counts for the tested instances are 68 vs 140 (GAP), 90 vs 132 (MkCS), and 38 vs 120 (IP) for HUBO versus QUBO.
To reach fixed approximation-ratio thresholds (0.2–0.6), HUBO-QAOA uses 89.6–100% fewer CNOTs and 86.1–100% fewer RZ gates than QUBO-QAOA across all benchmarked sizes.
At equal QAOA depth HUBO produces better average objective values in all three problems, e.g., 12.1 vs 15.8 minutes walking time for the GAP instance.
The full cost unitary is implemented with only single- and two-qubit gates, so the method is directly compatible with current device constraints despite higher-order Hamiltonian terms.

Reading between the lines

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

The exponential qubit saving is structural and does not depend on QAOA; it should transfer to any algorithm that needs to encode m-valued variables, including quantum annealing or imaginary-time evolution.
The ≥89.6% CNOT figure combines a deterministic per-layer advantage with an empirical convergence-rate comparison; under a different optimizer or with fixed instead of per-encoding-tuned penalty weights, the total-gate saving could be smaller, though still nonzero.
A natural stress test is to apply the same construction to problems with inequality constraints involving more than two variables, where QUBO typically requires extra slack variables; HUBO's direct projector penalties may avoid that overhead entirely.
The Gray-code compilation implies a general rule: any set of Z-terms sharing the same qubit block can be implemented with one CNOT per new subset, so the reported per-layer counts are lower bounds for any Hamiltonian with similar locality.

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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

Two things you should know. First, the deterministic part of this paper is solid: mapping n variables with m values to n·ceil(log2 m) qubits instead of n·m, and compiling each QAOA layer to explicit CNOT counts, gives real per-layer savings (roughly 32–68% on the three example instances). The coefficient construction via Walsh–Hadamard transform is clean, the O(n^2 m^2) per-layer gate scaling is derived, and the Gray-code circuit optimization is a sensible, if known, tool. They ship PyHUBO, which is a useful artifact. Second, the headline 'decreases total CNOT gate counts by at least 89.6%' does not come from those compiled circuits. It comes from the number of QAOA layers needed to reach a target approximation ratio, and Appendix D states they ran each encoding 100 times and picked the run with the lowest layer requirement. Minima over 100 runs can systematically favor whichever encoding has smaller run-to-run variance, and the Lagrange multipliers are tuned separately for each encoding. The stress-test note lands: the distribution of layer counts is never reported, so the 89.6% number is not robust. My guess is HUBO still genuinely needs fewer layers here—the fixed-depth averages in Figures 7, 10, and 13 look better—but the magnitude of the claimed savings is uncertain.

Other soft spots, in proportion. The benchmarks are tiny: n up to 5, m fixed at 4 (a power of two), so non-power-of-two m is never tested and the scaling plots have five points without error bars on the threshold measurements. The asymptotic gate scaling is the same class for both encodings, O(n^2 m^2); the advantage is constant-factor plus the exponential qubit reduction. The abstract could leave the impression that the gate-complexity class improves, which it does not. None of these are fatal, but they should be fixed before the 'practical alternative' framing is accepted at face value.

The authors are unusually transparent about their selection protocol, and the deterministic math is correct. This deserves peer review. The right referee will push for average/percentile layer counts, a pre-registered or at least fixed lambda selection, tests with m not equal to a power of two, and a careful separation of per-layer circuit facts from empirical QAOA performance. With those revisions this becomes a useful reference for anyone working on encoding comparisons in quantum optimization.

Referee Report

3 major / 5 minor

Summary. The paper proposes a systematic HUBO encoding for combinatorial optimization problems with n variables, each taking m values, using d = ceil(log2 m) qubits per variable instead of the n*m qubits of a one-hot QUBO encoding. The authors derive the Pauli expansion of the cost Hamiltonian via a Walsh-Hadamard transform (Eqs. 15-23), implement the diagonal cost unitary with Gray-code parity circuits using only CNOT and RZ gates, and prove O(n^2 m^2) per-layer scaling. They benchmark QUBO-QAOA and HUBO-QAOA on small Gate Assignment, Maximum k-Colorable Subgraph, and Integer Programming instances, reporting qubit counts, per-layer gate counts, and total gates needed to reach target approximation ratios. The central advertised result is that HUBO reduces qubit requirements and cuts total CNOT counts by at least 89.6% after compilation for all tested instances. An open-source package, PyHUBO, is released.

Significance. If the empirical claims hold, the paper offers a practical and broadly applicable alternative to QUBO for QAOA-style optimization: the qubit reduction from n*m to n*log m is clean and deterministic, the Pauli/Hadamard construction is parameter-free, and the Gray-code circuit synthesis is explicit and machine-checkable in principle. The per-layer deterministic gate counts in Table 2 show real, if modest, HUBO advantages (32-68% CNOT reduction for the three concrete instances), and the asymptotic scaling in Appendix C is useful. The 89.6-100% CNOT savings, however, are not a property of the encodings alone: they depend on empirical QAOA convergence, on best-of-100 run selection, and on per-encoding penalty tuning. Since that headline number is the paper's main selling point, its evidential basis needs to be substantially strengthened before the central claim can be accepted.

major comments (3)

[Appendix D / Figs. 8, 11, 14] The headline 'at least 89.6% CNOT reduction after compilation' is not supported by the deterministic per-layer counts in Table 2, which show only 32-68% reductions. The larger figure comes from the number of layers needed to reach target thresholds, and the protocol in Appendix D states that for these benchmarks the authors 'ran the QAOA algorithm 100 times and picked the QAOA experiment with the lowest layer requirements.' The distribution of required layers is not reported, and the Lagrange multipliers lambda are tuned iteratively per encoding without a fixed protocol. If the two encodings have different run-to-run variance, the minimum over 100 runs can systematically favor one encoding (here HUBO) even when typical performance is similar. Please report medians/quantiles, success probabilities as a function of layers, and use a symmetric, pre-specified lambda-selection rule. Without t
[Abstract and Sec. 2] The claim that HUBO 'exponentially reduces qubit requirements' is not accurate as stated. QUBO uses n*m qubits and HUBO uses n*ceil(log2 m) qubits, so the ratio is m/ceil(log2 m), which is polynomial in m. The correct statement is that the qubit count is reduced from linear to logarithmic in m. This wording appears in the abstract and is repeated in Sec. 2 and the Conclusion; it should be corrected to avoid overstating the advantage.
[Sec. 2.3 / Fig. 14] The paper states that QUBO-QAOA was 'not computationally feasible' at the n=5, m=4 IP instance, which is only 20 qubits, while QUBO-QAOA is reportedly run on a 16-qubit reduced instance. This is surprising and unexplained: statevector simulation of 20 qubits is routine in PennyLane. If the largest IP instance is excluded for QUBO, then the percentage savings quoted for IP (94.4-100%) apply only to the smaller instances; the text should state this explicitly and justify the claimed infeasibility.

minor comments (5)

[Sec. 1.1] The sentence 'without loss of generality, we only consider linear and quadratic COPs' is not literally WLOG; higher-order objectives are simply outside the scope. Please rephrase.
[Sec. 1.2.2 / Eq. (16)] The binary-to-decimal mapping is ambiguous when values are indexed from 1 to m. Since a d-bit string decodes to 0..2^d-1, the penalty in Eq. (16) may be off by one. Please define the offset explicitly (e.g., value index v = binary + 1 or v = binary).
[Appendix A] The stated complexity O(n^2 m^6) for computing quadratic HUBO coefficients overstates the cost of the Walsh-Hadamard transform. H^{⊗d} can be applied to a vector of length 2^d in O(d 2^d) time, so the per-pair cost is O(d^2 2^{2d}) = O(m^2 log^2 m), giving O(n^2 m^2 log^2 m) overall. Please correct or clarify the complexity model.
[Appendix A, Eq. (28)] Typo: 'co + c1' should be 'c0 + c1'.
[Table 2] Please state explicitly whether the reported QUBO gate counts include the one-hot penalty terms and any constraint penalties. This is important for reproducing the per-layer comparisons.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; HUBO construction is an exact transform of the classical objective and resource counts are explicit, parameter-independent circuit counts.

full rationale

The derivation chain is self-contained. The HUBO coefficients in Eq. (23) are obtained by substituting the Pauli expansion of projectors, Eqs. (18) and (21), into the objective Hamiltonian Eq. (15); no fitted or optimized constant is inserted into this step, and the same coefficients are used to build the circuits. The qubit-count comparison (n×m vs n⌈log2 m⌉) and the per-layer CNOT/RZ counts in Table 2 follow from the explicit encodings and circuit constructions, and the asymptotic scalings in App. C are derived for both encodings from the same worst-case coefficient counts. The Gray-code circuit optimization is attributed to external results [72,74], not to a self-citation, and the only self-reference, the PyHUBO package [31], is a software artifact rather than a load-bearing premise. The concern raised by a skeptic—that the 89.6–100% savings figures rest on 'the QAOA experiment with the lowest layer requirements' (App. D) and per-encoding lambda tuning—is an evidence-quality/statistical-validity issue about the representativeness of best-of-100 selection, not a circular reduction: the resource counts are not equal by construction to a fitted parameter, and the concrete per-layer counts independently support a resource advantage. No circular step is present.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central resource claims do not depend on exotic new physics; all ingredients are standard linear algebra and circuit synthesis. The main fitted ingredients are penalty multipliers, QAOA angles, and the choice of target approximation ratios. No new particles, forces, or conserved quantities are introduced.

free parameters (3)

Penalty multipliers lambda = not reported (tuned per instance)
Sec. 1.3: 'The Lagrange multipliers lambda ... are chosen iteratively before running the QAOA benchmarking. Only after identifying lambda values that ensure ... feasible solutions ... do we proceed.' The selected values are not listed, and they affect feasibility and gate counts because penalty terms add circuit terms.
QAOA variational parameters gamma, beta = not reported (optimized per instance and layer)
Sec. 1.3 uses gradient descent with Zhou initialization. The threshold-to-convergence gate counts in Figs. 8, 11, 14 depend on these optimized parameters.
Target approximation ratios for scaling benchmarks = 0.50 (GAP), 0.20 (MkCS), 0.30 (IP)
Chosen per problem class in Secs. 2.1-2.3; different thresholds change the reported gate-count savings, so they are effective free degrees of freedom in the comparison.

assumptions (4)

domain assumption Polynomial objective of a COP can be written as a sum of linear and quadratic terms (Eq. 1) without loss of generality.
The framework is derived for linear+quadratic objectives; higher-order objectives are excluded from benchmarking.
domain assumption Constraints can be enforced by adding lambda times a violation indicator to the Hamiltonian, with lambda large enough that the ground state is feasible (Eqs. 3-4).
Used to build both QUBO and HUBO Hamiltonians; in QAOA, finite lambda only approximately enforces feasibility, and the authors tune lambda empirically instead of proving a bound.
standard math Walsh-Hadamard expansion of projectors (Eq. 18) and the Gray-code parity circuit of Welch et al. [74] implement exp(i gamma J_T Z_T) using one CNOT per Gray-code edge.
Appendix A derives the expansion; Appendix B cites [74] for optimality of the Gray-code circuit. Both are standard but load-bearing for the CNOT count claims.
domain assumption Sampling 10,000 bitstrings gives an approximation ratio within 2% with 99% probability (Hoeffding), and averaging 100 independent QAOA optimizations is representative.
Appendix D. Relies on Monte Carlo sampling and ignores optimizer failure modes; the best-of-100 selection for thresholds is a further unmodeled choice.

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Cite this review

Pith. "Pith review of Resource-Efficient Quantum Optimization via Higher-Order Encoding." pith.science (2026). https://pith.science/paper/TPB2ACKN

@misc{pith2026251117545,
  author       = {Pith},
  title        = {Pith review of: Resource-Efficient Quantum Optimization via Higher-Order Encoding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TPB2ACKN}},
  note         = {Machine review of arXiv:2511.17545}
}

read the original abstract

Quantum approaches to combinatorial optimization problems (COPs) are often limited by the resource demands of Quadratic Unconstrained Binary Optimization (QUBO) encodings, which enlarge circuits through penalty terms and increase qubit and gate counts. We show that Higher-Order Unconstrained Binary Optimization (HUBO) enables a more resource-efficient formulation. Our method systematically constructs HUBO Hamiltonians and, compared to a QUBO formulation in benchmarks on Gate Assignment (GAP), Maximum k-Colorable Subgraph (MkCS), and Integer Programming (IP) problems, significantly reduces qubit requirements and decreases total CNOT gate counts by at least 89.6% for all tested instances. These results highlight HUBO as a practical alternative for quantum optimization on near-term devices. To promote adoption, we release an open-source Python library that automates HUBO model construction, extends beyond the examples presented in this work, and broadens access to resource-efficient quantum optimization.

Figures

Figures reproduced from arXiv: 2511.17545 by the authors.

**Figure 2.** Illustration of the GAP with m = 4 airportgates and n = 3 flights. Two of the flights have passengers who require a transfer connection, which is represented by the two-way symbol. A third flight has only arriving and departing passengers, illustrated by the luggage symbol. The goal is to assign airport-gates to flight such that the total passenger walking time is minimized. assigning flight i to airport-gate v … view at source ↗

**Figure 3.** Example instance for the MkCS problem: a [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Schematic representation of the QAOA al [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Figures from the paper (7 more)

**Figure 6.** Figure 6: Upper panel: Airport-Gate layout for the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 10.** Figure 10: Average approximation ratio achieved by QAOA as a function of total gate count for HUBO and QUBO encodings. Results are shown for the same graph with five vertices and nine edges as in [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: Scaling of quantum resource requirements for [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

**Figure 13.** Figure 13: Average approximation ratio versus total gate [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

**Figure 16.** Figure 16: Gray code traversal for optimal implementation of a 4-qubit HUBO Hamiltonian. Each vertex of the [PITH_FULL_IMAGE:figures/full_fig_p017_16.png]

**Figure 17.** Figure 17: Non-optimized implementation of the cost unitary for a three-qubit HUBO Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p017_17.png]

**Figure 18.** Figure 18: Optimal quantum circuit for implementing the cost unitary of a three-qubit HUBO Hamiltonian using [PITH_FULL_IMAGE:figures/full_fig_p018_18.png]

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