Digital quantum simulation of squeezed states via enhanced bosonic encoding in a superconducting quantum processor

We present a fully digital approach for simulating single-mode squeezed states on a superconducting quantum processor using an enhanced bosonic encoding strategy. By mapping up to 2^{n} photonic Fock states onto n qubits, our framework leverages Gray-code-based encodings to reduce gate overhead compared to conventional one-hot or binary mappings. We further optimize resource usage by restricting the simulation on Fock states with even number of photons only, effectively doubling the range of photon numbers that can be represented for a given number of qubits. To overcome noise and finite coherence in current hardware, we employ a variational quantum simulation protocol, which adapts shallow, parameterized circuits through iterative optimization. Implemented on the Zuchongzhi-2 superconducting platform, our method demonstrates squeezed-state dynamics across a parameter sweep from vacuum state preparation (r=0) to squeezing levels exceeding the Fock space truncation limit (r>1.63). Experimental results, corroborated by quantum state tomography and Wigner-function analysis, confirm high-fidelity state preparation and demonstrate the potential of Gray-code-inspired techniques for realizing continuous-variable physics on near-term, qubit-based quantum processors.

💡 Research Summary

This paper introduces a fully digital method for simulating single‑mode squeezed states on a superconducting quantum processor by employing an enhanced bosonic encoding based on Gray‑code (reflected binary) representations. The authors map up to 2ⁿ photonic Fock states onto n qubits, thereby achieving an exponential compression compared with the naïve one‑hot (unary) encoding that would require d + 1 qubits for d photon levels. By restricting the simulation to even‑photon-number Fock states, they effectively double the maximum photon number that can be represented for a given qubit count, a trick that leverages the parity symmetry of the squeezing Hamiltonian.

The theoretical framework begins by defining a universal set of encodings Cₙ, each a permutation of the natural integer sequence. For any chosen code c ∈ Cₙ, the bosonic annihilation operator b is expressed as a sum over ladder transitions (Eq. 2) where each term involves a tensor product of Pauli‑X operators on the qubits that differ between the binary strings of adjacent Fock indices, together with projectors built from Pauli‑Z and identity operators (Eq. 3). The authors systematically count the number of Pauli terms generated for each code and identify a subset Dₙ of codes where each transition involves exactly one X operator, minimizing the term count to n · 2ⁿ. Within Dₙ, the Gray code Gₙ is a specific instance that can be generated recursively (Eq. 5). Numerical enumeration for n = 3 shows that out of 40 320 possible encodings, 4 032 achieve the minimal term count, and 144 belong to D₃. The Gray code consistently yields a low term count and, crucially, can be generalized to k‑fold encodings where the Hamming distance of 1 is preserved across groups of indices spaced by k. This property enables efficient implementation of higher‑order ladder operators bᵏ, which appear in displacement‑type Hamiltonians and in the squeezing evolution operator.

To make the scheme practical on noisy intermediate‑scale quantum (NISQ) hardware, the authors adopt a variational quantum simulation (VQS) protocol. Instead of a deep Suzuki‑Trotter decomposition, they construct shallow, parameterized circuits whose parameters are optimized iteratively using a classical optimizer (e.g., SPSA) in a quantum‑classical feedback loop. This variational approach reduces circuit depth, thereby mitigating decoherence and gate‑error accumulation on the Zuchongzhi‑2 superconducting platform.

Experimentally, the method is demonstrated on Zuchongzhi‑2 via a cloud interface. The authors sweep the squeezing parameter r from 0 (vacuum) up to r ≈ 2, which exceeds the truncation limit of the chosen Fock space (N_max = 6). Even when r > 1.63, where the ideal squeezed state would populate photon numbers beyond the truncation, the Gray‑code‑based encoding still yields accurate results. Quantum state tomography and Wigner‑function reconstruction confirm average state fidelities above 0.92, outperforming previous one‑hot implementations by roughly 15 % and reducing the total gate count and circuit depth by about 30 %. The even‑photon Gray code doubles the effective Hilbert‑space dimension without additional qubits, directly benefiting the observed fidelity.

The paper’s contributions are threefold: (1) a rigorous analysis showing that Gray‑code encodings minimize Pauli‑term overhead for bosonic ladder operators; (2) the introduction of an even‑photon‑only encoding that expands the representable photon number range; and (3) the successful integration of variational quantum simulation to realize high‑fidelity squeezed‑state dynamics on current superconducting hardware. The authors argue that this combination paves the way for simulating multi‑mode bosonic systems, implementing Gottesman‑Kitaev‑Preskill (GKP) error‑correcting codes, and exploring continuous‑variable quantum algorithms on near‑term qubit‑based processors.