Efficient implementation of single particle Hamiltonians in exponentially reduced qubit space

Current and near-term quantum hardware is constrained by limited qubit counts, circuit depth, and the high cost of repeated measurements. We address these challenges for solid state Hamiltonians by introducing a logarithmic-qubit encoding that maps a system with $N$ physical sites onto only $\lceil \log_2 N \rceil$ qubits while maintaining a clear correspondence with the underlying physical model. Within this reduced register, we construct a compatible variational circuit and a Gray-code-inspired measurement strategy whose number of global settings grows only logarithmically with system size. To quantify the overall hardware load, we introduce a volumetric efficiency metric that combines the number of qubit, circuit depth, and the number of measurement settings into a single measure, expressing the overall computation costs. Using this metric, we show that the total space-time-sampling volume required in a variational loop can be reduced dramatically from $N^2$ to $(logN)^3$ for hardware efficient ansatz, allowing an exponential reduction in time and size of the quantum hardware. These results demonstrate that large, structured solid-state Hamiltonians can be simulated on substantially smaller quantum registers with controlled sampling overhead and manageable circuit complexity, extending the reach of variational quantum algorithms on near-term devices.

💡 Research Summary

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The paper tackles three major bottlenecks of near‑term quantum simulation—limited qubit count, circuit depth, and the overhead of repeated measurements—by focusing on single‑particle (or single‑excitation) Hamiltonians that arise in tight‑binding models of solids. While a system with N lattice sites formally requires an N‑dimensional Hilbert space, the conventional quantum encoding uses N qubits, yielding a 2^N‑dimensional space in which only N basis states are physically relevant. The authors propose a logarithmic‑qubit encoding that maps each physical site k (0 ≤ k < N) to the binary representation b(k) = (b₁,…,b_n) and stores it in a register of n = ⌈log₂ N⌉ qubits. In the ideal case N = 2ⁿ the mapping is one‑to‑one; when N lies between two powers of two, the register contains extra basis states that do not correspond to any physical site.

Two strategies are presented to keep the variational algorithm confined to the physical subspace. The first introduces an extended Hamiltonian H_ex = H₀ + C_p ∑_{i=N+1}^{2ⁿ}(1−Z_i)/2, where a large penalty constant C_p energetically suppresses population of the unphysical states. The second restricts the ansatz itself so that its reachable states are automatically limited to the N‑dimensional subspace. Both approaches are shown to work in numerical experiments.

The variational ansatz builds on the previously introduced Single‑Excitation Subspace (SES) circuit. Starting from the localized excitation |e₁⟩ = X₁|0…0⟩, a sequence of two‑qubit gates A_{j,j+1}(β_j,γ_j) propagates the excitation across the register. Each A gate decomposes into three CNOTs and a set of single‑qubit rotations (R_y, R_z), allowing the parameters β_j and γ_j to control both amplitude and phase of the excitation on site j. After applying N−1 such gates, the trial state becomes |ψ(θ)⟩ = ∑_{k=1}^N α_k|e_k⟩, where the complex coefficients α_k are variational parameters. Crucially, this construction works unchanged when the register size is logarithmic, because the binary encoding guarantees that each computational basis state corresponds to a unique physical site.

The Hamiltonian itself is the standard tight‑binding form H = ∑k h{kk}|k⟩⟨k| + ∑{j≠k} h{jk}|j⟩⟨k|. When expressed in Pauli operators, diagonal terms become (1−Z_k)/2 and hopping terms become combinations of X_jX_k, Y_jY_k, Y_jX_k, and X_jY_k. To evaluate ⟨ψ(θ)|H|ψ(θ)⟩, the authors devise a measurement scheme inspired by Gray codes. By ordering measurement bases according to a Gray‑code sequence, successive bases differ by only a single qubit rotation, which reduces the total number of distinct global measurement settings from O(N) (as in naïve approaches) to O(log N). This dramatically cuts the number of required circuit re‑preparations and statistical samples.

To quantify overall hardware demand, the paper introduces a “volumetric efficiency” metric V = (qubits) × (circuit depth) × (number of measurement settings). For a conventional SES implementation V scales as N² (since qubits ≈ N, depth ≈ N, and settings ≈ N). In the logarithmic scheme, qubits scale as log N, depth as log N (the number of A‑gates is proportional to the number of physical sites but each gate acts on a pair of qubits, so the depth grows only logarithmically when parallelized), and measurement settings also scale as log N, giving V ∝ (log N)³. Numerical simulations confirm that the expectation‑value estimates retain the same accuracy while the volumetric cost is reduced by orders of magnitude for realistic system sizes (e.g., N = 1024 → n = 10 qubits, V reduced from ~10⁶ to ~10³).

The authors also discuss practical considerations such as error mitigation, choice of the penalty constant C_p, and the impact of hardware noise on the shallow ansatz. They demonstrate that even with modest gate fidelities, the reduced qubit count and shallow depth make the approach viable on current superconducting and trapped‑ion platforms.

In summary, the paper provides a complete framework—encoding, ansatz design, measurement strategy, and a quantitative resource metric—that enables the simulation of large single‑particle Hamiltonians on near‑term quantum devices using exponentially fewer qubits. By converting the dominant N‑scaling of qubits, circuit depth, and measurement overhead into logarithmic scaling, the work opens a realistic pathway toward quantum‑enhanced calculations in solid‑state physics, quantum chemistry, and materials science, where tight‑binding models are ubiquitous.