Simple and efficient end-to-end quantum thermal and ground state preparation

Problem Statement

The preparation of quantum thermal and ground states is a fundamental prerequisite for simulating many-body physics, chemistry, and materials science on quantum computers. Existing approaches face a persistent trade-off between simplicity, efficiency, generality, and rigor. Variational algorithms require parameter tuning; adiabatic methods can be slow; and high-order dissipative algorithms (e.g., based on Lindblad dynamics) often require complex circuit structures, including controlled or time-reversed Hamiltonian evolutions and large numbers of ancilla qubits. These complexities make them challenging to implement on early fault-tolerant quantum devices. Conversely, simplified protocols often lack rigorous theoretical guarantees regarding mixing times or fixed-point accuracy. The goal is to develop an algorithm that is simple enough for early hardware yet admits rigorous end-to-end performance guarantees for physically relevant Hamiltonians.

Methodology

The authors propose a randomized system-bath interaction scheme that requires only forward time evolution under a system-bath Hamiltonian and a single reusable ancilla qubit. The protocol operates as follows:

1. Hamiltonian Setup: The total Hamiltonian is defined as $H_\alpha(t) = H + H_E + \alpha f(t) (A_S \otimes B_E + A_S^\dagger \otimes B_E^\dagger)$, where $H$ is the system Hamiltonian, $H_E = -\omega Z/2$ is the bath Hamiltonian depending on a random frequency $\omega$, and $A_S$ is a coupling operator randomly chosen from a set $\mathcal{A}$.
2. Bath Initialization: The ancilla qubit (bath) is initialized in a state $\rho_E \propto \exp(-\beta H_E)$. For ground state preparation ($\beta = \infty$), the bath is initialized in $|0\rangle\langle 0|$.
3. Interaction and Evolution: In each iteration, a coupling operator $A_S$ and a frequency $\omega$ are sampled from prescribed distributions. The system and bath evolve forward under the time-dependent Hamiltonian $H_\alpha(t)$ with a smooth temporal envelope $f(t)$ (chosen as a Gaussian function).
4. Trace and Reset: The ancilla qubit is traced out and reset to $\rho_E$. This process defines a quantum channel $\Phi$ acting on the system.
5. Iteration: The channel $\Phi$ is iterated $\tau_{mix}$ times, where $\tau_{mix}$ is the mixing time required to reach the fixed point.

The implementation utilizes a second-order Trotter decomposition to simulate the time-dependent evolution, avoiding the need for backward Hamiltonian evolution or complex control logic.

Key Contributions

• Simple Architecture: The algorithm requires only a single ancilla qubit and forward Hamiltonian evolution, making it well-suited for early fault-tolerant and hybrid analog-digital devices.
• Rigorous Fixed-Point Guarantees: The authors prove that by carefully designing the bath state, coupling operators, and sampling distributions, the fixed point of the quantum channel $\Phi$ can be made arbitrarily close to the desired thermal or ground state.
• Mixing Time Analysis: The paper establishes rigorous polynomial mixing-time bounds for several classes of physically relevant Hamiltonians, resolving the tension between algorithmic simplicity and theoretical rigor.
• End-to-End Efficiency: By combining fixed-point error bounds with mixing time results, the authors provide a complete complexity analysis showing that the total runtime scales polynomially with system size and inverse precision for specific models.

Results

The paper presents three main theorems regarding the efficiency and correctness of the proposed algorithm:

1. Fixed-Point Approximation (Theorem 1): For any target precision $\epsilon$, there exist parameter choices such that the trace distance between the fixed point of the channel and the target state is at most $\epsilon$. The total Hamiltonian evolution time is bounded by a polynomial in the mixing time, inverse temperature (or spectral gap), and $1/\epsilon$.
2. Thermal State Mixing (Theorem 2): For thermal state preparation of non-interacting spin systems, quadratic fermionic systems (at constant $\beta$), and local commuting spin systems (at high temperature), the mixing time scales as $\text{poly}(N, 1/\epsilon)$.
3. Ground State Mixing (Theorem 3): For ground state preparation of non-interacting spin systems and quadratic fermionic systems, the mixing time also scales as $\text{poly}(N, 1/\epsilon)$.

The analysis relies on showing that the discrete quantum channel $\Phi$ approximates a continuous-time Lindblad dynamics. A critical technical contribution is demonstrating that the mixing time remains bounded (independent of the filter width parameter $\sigma$) even as $\sigma$ is increased to reduce the fixed-point error. This is achieved by designing the interaction to allow large energy transitions while keeping the perturbation to the approximate fixed point small.

Significance and Claims

The authors claim this work provides the first end-to-end efficiency guarantee within the system-bath interaction framework for both thermal and ground state preparation. Specifically:

• It is the first study to establish rigorous mixing-time guarantees for both thermal and ground states in this framework, laying a theoretical foundation for empirical results observed in related studies.
• The algorithm avoids the complexity of high-order Lindblad simulations (which often require time-reversed evolution and many ancillas) while maintaining rigorous performance guarantees.
• The protocol serves as a benchmark for dissipative engineering and co-design with error correction, as it tests the hardware's ability to support long-time evolution toward a stable fixed point using minimal circuitry.
• The work suggests that the weak system-bath interaction framework can be viewed as an approximate, low-order approach to Lindblad simulation, offering a practical path for early fault-tolerant devices.

The paper acknowledges that while the current implementation uses a second-order Trotterization (which may lead to asymptotic slowdowns compared to higher-order methods), the simplicity of the architecture is a primary advantage for near-term devices. It also notes that future improvements could involve higher-order Trotterization if backward evolution becomes efficient on specific hardware.