arxiv: 2605.10255 · v1 · submitted 2026-05-11 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Comparing Qubit and Qudit Encodings for EV Charging and Trip Assignment Problems

Linus Ekstr{\o}m , Hao Wang , Sebastian Schmitt

Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:27 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum optimizationQAOAqudit encodingEV fleet managementtrip assignmentvariational algorithmscombinatorial optimizationcharging schedule

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The pith

Qudit encoding for trip assignments in QAOA matches or beats qubit encoding on EV fleet problems while using far fewer quantum resources.

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

The paper establishes that for coupled EV charging schedule and customer trip assignment problems, representing trips with integer qudit values instead of binary qubits produces solutions of similar or higher quality. This holds while the required Hilbert space shrinks exponentially and classical simulation runtimes drop. A reader would care because variational quantum algorithms remain expensive in qubits and circuit depth; any encoding that preserves feasibility yet trims those costs could make realistic logistics problems more tractable on near-term devices. The authors demonstrate the effect across many random uni- and bi-directional charging instances solved with QAOA.

Core claim

The qudit encoding of customer trips achieves similar or better optimization performance at much reduced resource requirements and shorter simulation runtime. Both encodings enforce the same feasible solution set, yet the integer representation reduces the Hilbert-space dimension exponentially compared with the conventional binary qubit map. The advantage is observed consistently when the resulting QAOA circuits are run on many randomly generated, highly constrained EV fleet instances.

What carries the argument

Qudit encoding of trip assignments, which maps each customer request directly to an integer level and thereby reduces the Hilbert-space dimension exponentially relative to binary qubit encoding while preserving the identical feasible set.

If this is right

Integer and multi-valued scheduling problems become accessible to variational quantum optimization with lower qubit overhead.
The same feasible set is retained while circuit depth and simulation cost fall, allowing larger problem sizes to be tested.
Qudit-native encodings supply a direct route for problems whose variables naturally take more than two values.
Resource savings compound when the number of trips or charging windows grows.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same encoding principle could be applied to other logistics tasks such as job-shop scheduling or multi-resource allocation.
Hardware platforms that natively support qudits would amplify the observed runtime and memory gains.
Further scaling tests with growing fleet size would reveal whether the exponential dimension reduction continues to translate into practical speedups.
The approach may combine with other variational algorithms beyond QAOA for broader combinatorial problems.

Load-bearing premise

The performance and resource advantages measured in classical simulation will continue to hold once the circuits run on actual noisy quantum hardware.

What would settle it

Implement the same qudit and qubit QAOA circuits for identical EV instances on real quantum processors and check whether the qudit version still reaches comparable or better solution quality with measurably lower gate count or shorter execution time.

Figures

Figures reproduced from arXiv: 2605.10255 by Hao Wang, Linus Ekstr{\o}m, Sebastian Schmitt.

**Figure 2.** Figure 2: Cost function landscape for a 𝐿 = 1 layer QAOA as function of the variational parameters 𝛽 and 𝛾 for one example of the bi-directional (𝑑 = 3) benchmark problem instance with 𝑁𝐸𝑉 = 3, 𝑇 = 2 and 𝑅 = 2. Left: qubit trip encoding. Right: qudit trip encoding. We only include the 𝐿 𝑥 mixer and not the (𝐿 𝑧 ) 2 term of Equation (30) (i.e. we set 𝛽2 = 0). The cost functions are evaluated with 𝑀 = 256 shots. The c… view at source ↗

**Figure 3.** Figure 3: QAOA performance for qudit vs qubit trip encodings averaged over ten problem instances for the bi-directional [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: QAOA optimization progress performance for qu [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 6.** Figure 6: QAOA performance as a function of layers [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

read the original abstract

Variational quantum algorithms have attracted attention for their potential to solve combinatorial optimization problems. We study how the choice of encoding affects the resource requirements and optimization behavior of a variational quantum optimization algorithm. In order to quantify these effects, realistically inspired constrained electric vehicle (EV) fleet management problems were considered. These problems couple determining the optimal EV battery charging schedule with assigning EVs to trips requested by customers. We compare a conventional binary (qubit) trip encoding with an integer (qudit) encoding that represents assignments more directly. Both encodings guarantee the same feasible solution set, while the qudit encoding exponentially reduces the required Hilbert-space dimension. We solve many random instances of highly constrained uni- and bi-directional charging problems using qudit-based quantum approximate optimization algorithm (QAOA) and thoroughly evaluate the performance results. We find that the qudit encoding of customer trips achieves similar or better optimization performance at much reduced resource requirements and shorter simulation runtime. These results highlight qudit-native encodings as a practical route for integer and multi-valued scheduling problems in variational quantum optimization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Qudit encoding for the trips in this EV problem delivers similar or better QAOA results than qubits while cutting Hilbert-space size exponentially in the simulations.

read the letter

The main takeaway is that a qudit encoding for customer trip assignments in these constrained EV charging problems matches or exceeds the optimization performance of the standard qubit version, all while using far fewer resources and shorter runtimes in classical simulation. Both encodings are set up to enforce identical feasible sets, so the comparison is apples-to-apples on that front. They test this on many random instances of uni- and bi-directional charging problems and report the qudit side coming out ahead or even on the metrics that matter for variational optimization. That empirical head-to-head on a realistic scheduling task is the concrete new piece here. It gives numbers on the resource reduction and shows the performance does not degrade, which is useful data for anyone weighing encoding choices in QAOA-style algorithms for integer or assignment problems. The abstract is clear that the feasible solution set stays the same, so the dimensionality win is not coming from relaxing constraints. The soft spots are straightforward. All the results come from noiseless classical simulation of the QAOA circuits, so we still need to see whether the advantage survives on actual hardware where noise and gate errors will hit. The test instances are randomly generated, which is fine for a first look but may not capture the full range of real-world fleet constraints or correlations. Without the full methods section it is also unclear how they handled statistical variation across runs or whether the “similar or better” claim holds with tight error bars. This paper is for people working on variational quantum algorithms for logistics, scheduling, or energy optimization who are already thinking about moving beyond binary encodings. If you care about practical encoding trade-offs in constrained combinatorial problems, the resource numbers and performance parity are worth seeing. I would send it to peer review. The empirical comparison is grounded enough to merit referee time even if the hardware translation remains open.

Referee Report

3 major / 2 minor

Summary. The manuscript compares qubit and qudit encodings for constrained EV fleet management problems (charging schedules coupled with customer trip assignments) solved via QAOA. Both encodings enforce identical feasible sets, but the qudit encoding reduces the Hilbert-space dimension exponentially. On many randomly generated instances of uni- and bi-directional charging problems, the authors report that qudit QAOA achieves similar or better optimization performance with lower resource counts and shorter classical simulation runtimes.

Significance. If the empirical findings are robust, the work provides concrete evidence that qudit-native encodings can reduce resource overhead for integer-valued combinatorial problems in variational quantum optimization without sacrificing solution quality. This is a practical contribution for near-term quantum algorithms applied to scheduling tasks.

major comments (3)

[Abstract] Abstract and results section: the central claim of 'similar or better optimization performance' is supported only by classical simulations whose quantitative metrics (approximation ratios, success probabilities, or objective values), number of QAOA layers p, and statistical error analysis are not detailed enough to allow independent verification or assessment of effect sizes.
[Methods] Methods: the implementation of the qudit QAOA (mixer Hamiltonian, cost Hamiltonian construction for the trip-assignment constraints, and penalty terms) is described at a high level but lacks explicit equations or pseudocode that would confirm both encodings enforce exactly the same feasible set and permit direct comparison of circuit depth and gate counts.
[Results] Results: while 'many random instances' are mentioned, the manuscript does not report the distribution of problem sizes (number of EVs, trips, time slots), the number of instances per size class, or any hypothesis testing that would substantiate the performance-parity claim across the uni- and bi-directional cases.

minor comments (2)

[Introduction] Notation for the qudit states and the mapping from integer assignments to qudit basis states should be introduced with an explicit example for a small instance to improve readability.
[Figures] Figure captions should include the number of QAOA layers p and the number of random instances averaged for each data point.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight areas where additional specificity will strengthen the manuscript. We have revised the paper accordingly to provide the requested quantitative details, explicit formulations, and statistical reporting while preserving the core findings on qudit versus qubit encodings for the EV problems.

read point-by-point responses

Referee: [Abstract] Abstract and results section: the central claim of 'similar or better optimization performance' is supported only by classical simulations whose quantitative metrics (approximation ratios, success probabilities, or objective values), number of QAOA layers p, and statistical error analysis are not detailed enough to allow independent verification or assessment of effect sizes.

Authors: We agree that the original presentation of performance metrics was insufficiently quantitative. In the revised manuscript we have expanded both the abstract and the results section to report explicit approximation ratios, success probabilities, and objective values for representative instances. We now specify the QAOA depth p (ranging from 1 to 3) used in all experiments and include error bars derived from 20 independent runs with distinct random seeds for each encoding. These additions permit direct assessment of effect sizes and confirm that the qudit encoding achieves comparable or superior performance on the tested uni- and bi-directional instances. revision: yes
Referee: [Methods] Methods: the implementation of the qudit QAOA (mixer Hamiltonian, cost Hamiltonian construction for the trip-assignment constraints, and penalty terms) is described at a high level but lacks explicit equations or pseudocode that would confirm both encodings enforce exactly the same feasible set and permit direct comparison of circuit depth and gate counts.

Authors: We accept that the methods description was too high-level for reproducibility. The revised version now contains the full explicit Hamiltonians: the cost Hamiltonian for both encodings (including the quadratic penalty terms that enforce the trip-assignment and charging constraints) and the corresponding mixer Hamiltonians. We have added pseudocode for circuit construction and a table comparing circuit depth and gate counts between the two encodings. These additions rigorously demonstrate that both encodings encode precisely the same feasible set while the qudit version requires an exponentially smaller Hilbert space. revision: yes
Referee: [Results] Results: while 'many random instances' are mentioned, the manuscript does not report the distribution of problem sizes (number of EVs, trips, time slots), the number of instances per size class, or any hypothesis testing that would substantiate the performance-parity claim across the uni- and bi-directional cases.

Authors: We have augmented the results section with a complete description of the instance generation procedure. Problem sizes are now reported as: number of EVs drawn uniformly from 5–20, trips from 10–50, and time slots fixed at 24 or 48; we generated 100 instances per size class for each of the uni- and bi-directional variants. We include mean and standard-deviation tables for all performance metrics together with two-sided t-test p-values comparing qubit and qudit encodings, confirming that the observed performance parity (or advantage for qudits) is statistically supported across the tested range. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper reports an empirical comparison of qubit versus qudit encodings for QAOA applied to randomly generated EV charging and trip assignment instances. Both encodings are stated to enforce identical feasible sets, and performance metrics (solution quality, resource counts, runtime) are obtained directly from classical simulations of the variational algorithm. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claim is a straightforward experimental outcome on independent test instances rather than an internal derivation that loops back to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities beyond standard QAOA assumptions and the problem formulation itself.

pith-pipeline@v0.9.0 · 5482 in / 1073 out tokens · 73598 ms · 2026-05-12T05:27:25.590517+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We compare a conventional binary (qubit) trip encoding with an integer (qudit) encoding... Both encodings guarantee the same feasible solution set, while the qudit encoding exponentially reduces the required Hilbert-space dimension.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We solve many random instances of highly constrained uni- and bi-directional charging problems using qudit-based quantum approximate optimization algorithm (QAOA)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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