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

This paper demonstrates a high-fidelity, fully digital simulation of single-mode squeezed states on the Zuchongzhi-2 superconducting processor by employing an enhanced Gray-code-based bosonic encoding and a variational protocol to efficiently map photonic Fock states onto qubits while mitigating hardware noise.

Original authors: Hengyue Li, Yusheng Yang, Zhe-Hui Wang, Shuxin Xie, Zilong Zha, Hantao Sun, Jie Chen, Jian Sun, Shenggang Ying

Published 2026-02-04

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CC BY 4.0

Original authors: Hengyue Li, Yusheng Yang, Zhe-Hui Wang, Shuxin Xie, Zilong Zha, Hantao Sun, Jie Chen, Jian Sun, Shenggang Ying

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build a digital video game that simulates a very wavy, fluid-like world (like light or sound waves). The problem is, your game console (the quantum computer) only understands "on/off" switches, like light switches in a house. It doesn't natively understand smooth waves.

This paper is about a clever new way to translate those smooth, wavy "bosonic" physics into the "on/off" language of a quantum computer, and then successfully playing that game on a real machine.

Here is the breakdown of their approach using simple analogies:

1. The Problem: Packing Too Many Things into Too Few Boxes

In physics, light is made of particles called photons. You can have 0 photons, 1 photon, 2 photons, and so on. To simulate this on a computer, you need to map these numbers to "qubits" (the computer's switches).

The Old Way (One-Hot Encoding): Imagine you have a row of light switches. To show "3 photons," you turn on the 4th switch and leave the rest off. To show "100 photons," you need 101 switches. This is very wasteful. If you want to simulate a lot of photons, you run out of switches immediately.
The New Way (Gray Code): The authors used a special "zipper" code called a Gray code. Think of it like a combination lock where you only have to turn one dial to go from one number to the next. This allows them to pack many more photon numbers into the same number of switches.
- The Result: With just 2 switches (qubits), the old way could only show up to 1 photon. Their new method could show up to 3.

2. The Secret Trick: The "Even-Number" Shortcut

The authors noticed something special about the "squeezed states" they wanted to simulate. These are specific quantum states where the light is "squeezed" in one direction and stretched in another.

The Trick: In these specific states, the number of photons is always even (0, 2, 4, 6...). You never get an odd number like 1 or 3.
The Analogy: Imagine you are packing a suitcase, but you know for a fact you will only ever pack pairs of socks. You don't need to make space for single socks.
The Result: By ignoring all the odd numbers, they effectively doubled their capacity again. With just 2 switches, they could now simulate states with up to 6 photons (0, 2, 4, 6). This is a huge leap in efficiency.

3. The Engine: A "Smart" Approximation (Variational Simulation)

Simulating how these states change over time usually requires a very long, complex sequence of instructions (a deep circuit). But current quantum computers are like fragile glass houses; if the instructions are too long or complex, the noise in the machine breaks the simulation before it finishes.

The Solution: Instead of trying to build a perfect, long bridge, they built a short, flexible bridge that they could adjust on the fly. They used a method called Variational Quantum Simulation (VQS).
The Analogy: Imagine trying to walk a tightrope. A rigid, pre-made path might be too long and wobbly. Instead, they used a flexible rope and a guide (a classical computer) that constantly checks their balance and tells them how to adjust their steps (parameters) to stay on the path. This keeps the "walk" short enough to survive the noise but accurate enough to reach the destination.

4. The Test Drive: The Zuchongzhi-2 Processor

They took their new encoding method and their "smart" simulation engine and ran it on a real quantum computer called Zuchongzhi-2 (made by QuantumCTek).

What they did: They started with a "vacuum" (no photons) and "squeezed" it to create a state with up to 6 photons.
The Result: They checked the results using two methods:
1. State Tomography: Like taking a 3D X-ray of the final state to see if it looks like the math predicted.
2. Wigner Function: A visual map that shows the "shape" of the quantum wave.
The Outcome: The results were very high quality. Even though the computer is noisy and they had to cut off the simulation at 6 photons (because they only had 2 switches), the shape of the wave they created matched the theoretical prediction very closely.

Summary

The paper claims that by using a special "Gray code" packing method and a trick to only count even numbers, they can simulate complex light physics on a small quantum computer much more efficiently than before. They proved this works on a real machine by successfully creating and measuring a "squeezed" light state, showing that digital quantum computers can handle continuous, wave-like physics even with limited hardware.

Technical Summary: Digital Quantum Simulation of Squeezed States via Enhanced Bosonic Encoding

Problem Statement
Simulating continuous-variable (CV) bosonic systems, such as photonic modes, on digital, qubit-based quantum hardware requires mapping infinite-dimensional Fock spaces onto finite qubit registers. Conventional encoding methods, such as one-hot (unary) encoding, incur linear resource overhead, requiring $d+1$ qubits to represent $d$ photon levels, which becomes impractical for large Hilbert spaces. While Gray codes (reflected binary codes) have been proposed to utilize the full $2^n$ computational basis of $n$ qubits, the efficiency of specific encoding schemes varies significantly depending on the target Hamiltonian terms (e.g., linear vs. higher-order bosonic operators). Furthermore, current superconducting quantum processors suffer from noise and limited coherence times, making deep circuit implementations (such as those required by strict Suzuki-Trotter decompositions) infeasible for accurate time evolution.

Methodology
The authors propose a fully digital approach combining an enhanced bosonic encoding strategy with a Variational Quantum Simulation (VQS) protocol.

Generalized Bosonic Encoding Framework:
- The paper establishes a framework mapping up to $2^n$ photonic Fock states onto $n$ qubits.
- It analyzes the "annihilation operator" $b$ and higher-order terms $b^k$ under various encoding permutations. The authors define a subset of codes $D_n$ where the Hamming distance between neighboring numbers is 1, minimizing the number of Pauli terms required to represent the operator.
- Gray Code Analysis: The study demonstrates that the Gray code ( $G_n$ ) is a specific instance within $D_n$ that is highly efficient for encoding operators $b^l$ where $l = 2^\xi$ (e.g., $b^2$ ). While $G_n$ is not optimal for all $k$ (e.g., $k=3$ ), it performs favorably for the specific squeezing Hamiltonian terms ( $b^2$ and $b^{\dagger 2}$ ) used in this work.
- Even-Photon Truncation: Exploiting the property that single-mode squeezed states contain only even-photon number components, the authors restrict the simulation to even-photon Fock states. This "even-photon" encoding effectively doubles the maximum representable photon number for a given $n$ (e.g., 2 qubits represent states up to 6 photons: $|0\rangle, |2\rangle, |4\rangle, |6\rangle$ ).
Variational Quantum Simulation (VQS):
- To overcome hardware noise and depth limitations, the authors employ a VQS protocol rather than standard Trotterization.
- Ansatz: They utilize a Variational Hamiltonian Ansatz (VHA), constructing the circuit by mimicking short Trotter-like steps of the original Hamiltonian. This allows for systematic improvement by increasing the number of layers ( $n_d$ ).
- Optimization: The time-dependent variational parameters are updated iteratively using the McLachlan variational principle to solve the equations of motion (EOM). This involves measuring the matrix $M$ and vector $V$ using ancilla-assisted circuits to determine the parameter gradients $\dot{\theta}(t)$ .
Experimental Implementation:
- The method was implemented on the Zuchongzhi-2 superconducting quantum processor via a cloud platform.
- The simulation focused on generating single-mode squeezed states from a vacuum initial state by evolving under the squeezing Hamiltonian $H \propto (b^2 - b^{\dagger 2})$ .
- The experiment swept the squeezing parameter $r$ from 0 (vacuum) to $r > 1.63$ , a regime where the average photon number exceeds the truncation limit of the 4-dimensional subspace (6 photons).

Key Results

Encoding Efficiency: The Gray code-based encoding, particularly when restricted to even-photon states, successfully represented Fock states up to 6 photons using only 2 qubits. The resulting Hamiltonian was decomposed into a manageable set of two-qubit Pauli strings.
Simulation Fidelity:
- Classical Simulation: The VQS simulation closely matched the exact theoretical solution (truncated to 6 photons) across the entire parameter sweep, with only minor deviations near $r \approx 1.25$ .
- Hardware Performance: On the Zuchongzhi-2 processor, the method achieved high-fidelity state preparation. Quantum State Tomography (QST) at $r=0.5$ yielded a state fidelity greater than 0.9 compared to the ideal truncated state.
- Wigner Function Analysis: Reconstructed Wigner functions from hardware data successfully reproduced the characteristic elongated shape and diagonal orientation of the squeezed state in phase space. While device noise and finite truncation introduced artifacts (such as negative "ring-like" regions), the core features of the squeezed state remained intact.
Scalability: The results demonstrated that the method can operate beyond the regime where the truncated Fock-space approximation typically breaks down due to saturation, provided the encoding is optimized.

Significance and Claims
The paper claims to demonstrate the feasibility of simulating continuous-variable (CV) physics on near-term, qubit-based digital quantum processors. The primary contributions are:

Enhanced Encoding: Validating that Gray-code-inspired techniques, specifically when combined with even-photon truncation, offer a resource-efficient pathway to simulate bosonic systems, reducing gate overhead compared to conventional mappings.
Robustness on NISQ Hardware: Showing that a Variational Quantum Simulation approach, utilizing shallow circuits and hybrid optimization, can effectively mitigate noise and coherence limitations to simulate non-trivial bosonic dynamics (squeezing) on current superconducting hardware.
Experimental Validation: Providing experimental evidence that digital quantum processors can emulate CV phenomena, bridging the gap between discrete-variable hardware and continuous-variable applications without requiring specialized CV hardware.

The authors conclude that their results establish a foundation for exploring a wider range of bosonic Hamiltonians on digital quantum processors using these encoding and variational strategies.