Quasiprobabilistic imaginary-time evolution on quantum computers

Technical Summary: Quasiprobabilistic Imaginary-Time Evolution on Quantum Computers

Problem Statement
Imaginary-time evolution (ITE), defined by the operator $e^{-\beta H}$, is a fundamental primitive for computing ground-state and thermal equilibrium properties of quantum many-body systems. While classical simulation of ITE is often intractable due to the exponential growth of the Hilbert space and the sign problem, existing quantum algorithms face significant barriers for near-term implementation. Many state-of-the-art approaches, such as those based on block-encoding, require fault-tolerant quantum computers with deep circuits and ancillary qubits, placing them beyond the reach of current noisy intermediate-scale quantum (NISQ) devices. Furthermore, current hardware is susceptible to noise, which degrades the performance of standard quantum algorithms. There is a need for algorithms that can perform ITE on current hardware without requiring full error correction, ancillary qubits, or complex multi-qubit controlled operations.

Methodology
The authors propose a new algorithm that implements ITE on quantum computers using quasiprobability decompositions (QPD), inspired by the error-mitigation technique of Probabilistic Error Cancellation (PEC). The core methodology involves the following steps:

1. Quasiprobability Decomposition of Linear Maps: The algorithm treats the ITE operator (and more generally, Hermiticity-preserving linear maps) as a linear combination of native operations available on a quantum device. Specifically, a target map $T$ is decomposed as $T(\rho) = \sum_i q_i B_i(\rho)$, where $B_i$ are native gates (including non-trace-preserving operations like projectors) and $q_i$ are real coefficients that may be negative.
2. Sampling and Post-Processing: The decomposition is converted into a probabilistic protocol. The coefficients are normalized to form a probability distribution $p_i = |q_i|/\gamma$, where $\gamma = \sum |q_i|$ is the sampling overhead. The algorithm probabilistically samples native circuits $B_i$ according to $p_i$. The measurement outcomes are classically post-processed, weighted by the sign of the coefficients ($\text{sgn}(q_i)$) and the factor $\gamma$, to reconstruct the expectation values.
3. Handling Non-Trace-Preserving Operations: Since ITE is non-trace-preserving, the algorithm estimates the rescaled expectation value $\langle A \rangle_{T(\rho)} = \text{tr}[A T(\rho)] / \text{tr}[T(\rho)]$. This is achieved by simultaneously estimating the numerator and denominator using indicator variables that account for post-selection outcomes (e.g., discarding runs where a projector fails).
4. Trotterization for Many-Body Systems: To apply this to an $n$-qubit Hamiltonian $H = \sum H_l$, the authors use a first-order Trotter-Suzuki approximation to decompose the global evolution into a sequence of local $k$-qubit operations. The QPD is computed for each local term separately, avoiding the intractable classical optimization required for the full $n$-qubit map.
5. Thermal Pure Quantum (TPQ) States: To estimate thermal expectation values, the algorithm prepares TPQ states. Instead of a Haar-random state (which is hard to prepare), the method applies a random Clifford unitary (a 3-design) to a computational basis state before performing the ITE. This allows the estimation of Gibbs state properties using pure states.

Key Contributions

• Novel Algorithm for NISQ Devices: The paper introduces a QPD-based algorithm for ITE that requires no ancillary qubits and avoids the implementation of complex multi-qubit controlled operations between data and ancilla qubits.
• Noise Resilience: By utilizing the native noisy gates of the device as the basis for the QPD, the algorithm is inherently noise-resilient. It does not require additional error-mitigation layers beyond the QPD post-processing itself, provided the noise channel is characterized.
• Scalability via Local Decomposition: The authors demonstrate that by combining QPD with Trotterization, the method scales to local Hamiltonians (e.g., Heisenberg models) by only requiring the decomposition of local terms, making it computationally feasible for systems larger than the few-qubit limit.
• Hardware Demonstration: The work includes a proof-of-principle demonstration on an IBM superconducting quantum processor (5-qubit ibm_manila) and extensive classical simulations.

Results

• 2-Qubit Heisenberg Model: The authors simulated and executed the algorithm on hardware to estimate the ground-state energy of a 2-qubit Heisenberg Hamiltonian. The hardware results showed agreement with the ideal ITE values, validating the noise-resilience of the approach. The authors noted that performance degraded slightly for higher Trotter steps, likely due to time-dependent noise drift affecting the accuracy of the noise characterization.
• Thermal Expectation Values: Numerical simulations were performed for 1D Heisenberg Hamiltonians on $n \in \{4, \dots, 8\}$ qubits. The algorithm successfully prepared TPQ states and estimated thermal expectation values for Pauli observables. The results showed good agreement with exact ITE values, with the error in the expectation value decaying rapidly as the number of qubits increased.
• Sampling Cost Analysis: Theoretical analysis confirms that the sampling cost scales as $O(\gamma^2/\epsilon^2)$, where $\gamma$ depends on the choice of the QPD basis. The authors showed that using a basis containing entangling gates (e.g., the Takagi basis) significantly reduces $\gamma$ compared to bases composed only of single-qubit operations (e.g., the EBL basis).

Significance and Claims
The paper claims that this approach offers a practical pathway for probing the physics of quantum many-body systems on current hardware. By leveraging quasiprobability decompositions, the method bypasses the need for fault tolerance and ancilla qubits, which are major bottlenecks for existing thermal state preparation algorithms. The authors emphasize that while the sampling cost can be high in the worst case, the rapid decay of TPQ state errors with system size makes the method particularly promising for near-term applications. The work positions itself as a "near-term friendly" addition to the quantum toolkit, extending the utility of error-mitigation techniques from noise correction to the implementation of non-unitary physical processes like imaginary-time evolution. The authors modestly note that further work is needed to optimize the choice of QPD bases for larger systems and to refine the complexity bounds for Trotterized non-trace-preserving evolution.