Interference Engineering for Quantum Imaginary-Time Evolution through Multiple Energy Shifts

Technical Summary: Interference Engineering for Quantum Imaginary-Time Evolution through Multiple Energy Shifts

Problem Statement
Imaginary-time evolution (ITE) is a fundamental tool for preparing ground states, thermal states, and simulating finite-temperature systems in quantum many-body physics. However, implementing ITE on quantum computers is challenging because the operator $e^{-H\tau}$ is non-unitary. Current approaches typically map ITE to a superposition of real-time evolutions. Two representative methods are Monte Carlo (MC) based approaches and continuous-variable (CV) assisted schemes.

• Monte Carlo limitations: These methods express ITE as a weighted superposition of real-time evolutions. The inherent sampling distribution possesses a long-time tail, necessitating frequent sampling of long real-time evolutions. This increases circuit depth and exacerbates noise accumulation on quantum hardware.
• Continuous-variable limitations: These methods rely on the postselection of an auxiliary continuous-variable state. The success probability is tightly coupled to the fidelity of the prepared state; achieving high fidelity often requires low success probability. Designing projection states that improve success rates without altering the evolution dynamics has been a practical bottleneck.
• The Energy Shift Paradox: In classical numerical algorithms, energy shifts are used to modify spectral properties. In standard ITE, adding a constant energy shift $b$ to the Hamiltonian ($H \to H+b$) only introduces an overall normalization factor ($e^{-b\tau}$), leaving the normalized final state unchanged. Consequently, energy shifts have rarely been utilized to optimize the evolution process itself. On quantum computers, however, where ITE is realized as a superposition of real-time evolutions, an energy shift manifests as a relative phase factor rather than just a normalization constant. This observation suggests that energy shifts can be engineered to manipulate interference patterns.

Methodology: Multi-Shift Quantum Imaginary-Time Evolution (MS-QITE)
The authors propose MS-QITE, a framework that replaces a single constant energy shift with a distribution of energy shifts characterized by a function $f(q)$. While the normalized imaginary-time evolution remains invariant, the distribution modifies the interference structure in the real-time representation.

The core formulation transforms the standard ITE operator into:
$$\int_{b_1}^{b_2} dq f(q) e^{-(H+q)\tau} \iff e^{-H\tau}$$
When mapped to real-time evolution, this introduces a weighting function $W(t) = \int dq f(q) e^{-iqt}$, which acts as a phase and amplitude modifier on the real-time evolution kernel. The paper explores two distinct implementations:

1. Monte Carlo Implementation:

   • The method samples pairs of real-time evolution durations $(t, t')$ from a probability distribution.
   • The multi-shift function $f(q)$ is chosen to define a specific $W(t)$. The authors select $W(t) = (a / (a + it))^k$.
   • This choice modifies the sampling probability density from $p(t) \sim 1/t^2$ to $p(t) \sim 1/t^{k+2}$.
   • By increasing the power $k$, the sampling distribution is concentrated within a shorter real-time window, reducing the required circuit depth.
   • Parameters are optimized to balance the suppression of long-time contributions against the introduction of excessive phase oscillations. The authors select $k=3$ and $a=k\tau$.

2. Continuous-Variable-Assisted Implementation:

   • ITE is realized by evolving an initial state jointly with a resource state and projecting the auxiliary mode onto a specific state $|\psi_q\rangle$.
   • The projection state's momentum-space wavefunction corresponds to the conjugate of the multi-shift function $W(t)$.
   • Instead of using a squeezed state (as in previous work), the authors propose a "step-function" projection state defined over a finite interval in position space: $|\psi_q\rangle = \frac{1}{\sqrt{L}} \int_{-L}^{0} dx |x\rangle$.
   • This corresponds to a constant $f(q)$ in the shift distribution. While a constant shift was deemed impractical in the MC approach due to phase oscillations, it is highly effective here.
   • The parameter $L$ is optimized (finding $L\tau \approx 1.256$) to maximize the postselection success probability.

Key Results
Numerical simulations were performed on transverse-field Ising models to validate the approach:

• Ground-State Preparation (MC):

  • Comparing the ground-state energy error with and without multi-shift, the MS-QITE method demonstrated lower errors and reduced fluctuations across different cutoff times ($t_m$).
  • The method successfully reduced the required real-time evolution duration, confirming the theoretical prediction that the sampling distribution is concentrated in a shorter time window.

• Thermal-State Preparation (CV-Assisted):

  • The authors compared the resource consumption (quantified by the squeezing factor) required to prepare thermal states with the same accuracy using the proposed step-function projection versus a traditional squeezed-state projection.
  • Results showed that the step-function state (MS-QITE) requires significantly fewer resources in the intermediate range of inverse temperatures.
  • The success probability of the algorithm stabilized as the photon-number truncation increased, while the trace distance to the ideal thermal state converged to zero.

Significance and Claims
The paper claims that MS-QITE provides a flexible algorithmic perspective for optimizing quantum imaginary-time evolution by exploiting interference engineering.

• Interference as a Resource: The work demonstrates that spectral transformations (energy shifts), traditionally viewed as trivial normalization factors, can be engineered as a resource to manipulate interference in real-time representations.
• Dual Utility: The framework offers complementary benefits: in hybrid quantum-classical schemes, it improves sampling efficiency by shortening effective evolution times; in fully quantum-assisted schemes, it enhances postselection efficiency by allowing projection onto larger subspaces.
• General Applicability: While demonstrated on Ising models, the authors suggest the approach is applicable to a broader class of quantum algorithms that rely on real-time evolution superpositions.

The paper concludes modestly, noting that future work may address optimal shift distributions, state-preparation overheads, and scalability to larger systems, without making definitive claims about immediate experimental deployment beyond the numerical verification provided.