Multiple-time Quantum Imaginary Time Evolution

This paper introduces the Multiple-Time Quantum Imaginary Time Evolution (MT-QITE) algorithm, which enhances ground state preparation fidelity and reduces measurement overhead compared to standard QITE by leveraging multiple imaginary time steps while maintaining determinism, independence from ad hoc ansatze, and parallelizability.

The Gist

Imagine you are trying to find the lowest point in a vast, foggy mountain range. This lowest point represents the "ground state" of a physical system—the most stable, lowest-energy configuration of atoms or particles. Finding this spot is crucial for understanding how materials behave, how chemical reactions happen, and for designing new drugs.

In the world of quantum computing, there is a method called Quantum Imaginary Time Evolution (QITE) that acts like a hiker who always steps downhill. Over time, this hiker naturally settles into the deepest valley (the ground state). However, the paper by Del Castillo, Granath, and van Nieuwenburg points out a major problem with the standard hiker: the path is expensive. To take each step, the hiker has to stop, measure the terrain, and do a lot of math. This "measurement budget" is like a limited supply of fuel; if you run out of fuel (measurements), you can't finish the journey.

The New Solution: The "Multiple-Time" Hiker (MT-QITE)

The authors introduce a new algorithm called MT-QITE (Multiple-Time Quantum Imaginary Time Evolution). Instead of just one hiker taking one type of step, imagine a team of hikers working together, or a single hiker who can try different step sizes simultaneously to find the most efficient path.

Here is how the paper explains the improvements using simple concepts:

1. The "Try It All" Strategy (Variational Flexibility)
In the old method (QITE), the hiker had to commit to a specific step size (time step) for the whole journey. If they picked a step that was too big, they might overshoot the valley floor. If it was too small, the journey would take forever.

• The MT-QITE Analogy: Imagine the hiker can now test several different step sizes at once without actually walking the whole path for each one. They calculate the outcome of taking a small step, a medium step, and a large step all at the same time using the same starting point. Then, they simply pick the one that leads to the lowest energy. This flexibility allows them to find a better "lowest point" (higher fidelity) without wasting extra fuel.

2. The Shared Map (Parallelization)
The old method was like a relay race where the second runner couldn't start until the first runner finished their entire leg and updated the map. This meant the hiker had to stop and measure the terrain again and again for every single step.

• The MT-QITE Analogy: MT-QITE is like a team of explorers sharing a single, high-resolution map. Because they all start from the same reference point, they can measure the terrain once and use that data to calculate the best moves for all the different step sizes simultaneously. This means they don't have to stop and measure as often. The paper claims this reduces the number of measurements (the "fuel") needed by a factor of 10 in some cases.

3. The "Symmetry" Shortcut
The paper notes that many physical systems have symmetry (like a mirror image). If you know the left side of the mountain looks like the right side, you don't need to measure both sides.

• The MT-QITE Analogy: Because the MT-QITE team shares a single map, they can easily use these symmetry shortcuts. If they measure one part of the terrain, they can mathematically deduce the rest without taking extra measurements. The old method couldn't do this as easily because the "map" kept changing after every single step.

What the Results Show

The authors tested this new method on four different "mountain ranges" (physical models):

• The Ising Model: A model of magnetic spins.
• The Heisenberg Model: Another magnetic model.
• The Hubbard Model: A model for electrons in materials.
• The H4 Chain: A small molecule made of four hydrogen atoms.

In all these tests, the MT-QITE method found the "lowest valley" (the ground state) much more accurately than the old method.

• Better Accuracy: In some cases, the new method was 10 to 100 times more accurate.
• Less Fuel: It required significantly fewer measurements (about 10 times fewer) to get that accuracy.
• No Guesswork: Unlike other methods that require the user to guess a "shape" for the solution (an ansatz), MT-QITE figures out the best path automatically at every step.

The Bottom Line

The paper concludes that MT-QITE is a more efficient, deterministic, and accurate way to find the ground states of quantum systems. It doesn't rely on luck (probabilistic methods) or pre-set guesses (ansatz). By allowing the algorithm to "try" multiple imaginary time steps at once using a shared reference state, it saves a massive amount of computational resources while delivering a better result.

The authors emphasize that this is currently a simulation on classical computers, but the method is designed to run on both current noisy quantum devices and future error-corrected quantum computers. They have made their code available for others to test.

Technical Summary

Technical Summary: Multiple-Time Quantum Imaginary Time Evolution (MT-QITE)

Problem Statement
Quantum Imaginary-Time Evolution (QITE) is a deterministic, ansatz-free method for preparing ground states on quantum hardware by approximating non-unitary imaginary-time evolution with unitary real-time evolution (RTE). However, standard QITE faces significant challenges regarding measurement overhead and computational cost, particularly for general Hamiltonians. The fidelity and efficiency of QITE are heavily dependent on the definition of local domains and Hamiltonian partitions. Furthermore, the standard algorithm updates the reference quantum state at every Trotter step, preventing the reuse of measurements across different Hamiltonian terms and limiting parallelization. While variational approaches like VQE exist, they rely on ad hoc ansatz selection, whereas QITE aims to determine the evolution operator optimally at each step without such assumptions.

Methodology
The authors introduce the Multiple-Time QITE (MT-QITE) algorithm, a heuristic extension of the standard QITE protocol. The core innovation lies in decoupling the time step size for different terms within a Hamiltonian partition.

1. Multiple-Time Ansatz: Instead of applying a single time step $\Delta \tau$ to all Hamiltonian terms in a Trotter step, MT-QITE assigns distinct time parameters ($t_1, t_2, \dots$) to each partition term ($\hat{h}[1], \hat{h}[2], \dots$). The evolved state is defined as:
   $$|\Psi'(t_1, t_2)\rangle = \frac{e^{-t_2\hat{h}[2]}e^{-t_1\hat{h}[1]}|\Phi(0)\rangle}{\|e^{-t_2\hat{h}[2]}e^{-t_1\hat{h}[1]}|\Phi(0)\rangle\|}$$
   This introduces additional variational degrees of freedom, allowing the algorithm to minimize the ground energy upper bound more effectively than the single-time constraint of standard QITE.

2. Shared Reference State: Unlike standard QITE, which updates the quantum state after processing each Hamiltonian term, MT-QITE uses a fixed reference state $|\Phi\rangle$ for the entire Trotter step. This allows the linear systems of equations (derived from tomography) to be solved for multiple time step sizes using the same set of measurements.

3. Parallelization and Symmetry: Because the reference state remains constant across all partition terms during a step, the routine is parallelizable. Furthermore, if the Hamiltonian possesses symmetries (e.g., site inversion), the algorithm can reuse measurements for symmetric terms without re-evaluating the quantum state, significantly reducing the measurement budget.

4. Quantum Chemistry Reformulation: The paper presents a reformulation of QITE equations specifically for quantum chemistry Hamiltonians. By utilizing anti-hermitian operators (such as those in Unitary Coupled-Cluster with Singles and Doubles, UCCSD) rather than general Pauli strings, the method ensures the conservation of particle number and spin. This approach reduces the dimension of the required linear system by approximately a factor of 8 and allows for the simultaneous measurement of up to 64 Pauli strings via mutually commuting sets.

Key Results
The authors benchmarked MT-QITE against the original QITE algorithm (with optimized time steps) on four physical systems using noiseless classical emulation:

• XXZ Heisenberg Model (8 qubits): At the phase transition ($J=1$), MT-QITE achieved a ground state fidelity improvement of one order of magnitude compared to QITE after 10 Trotter steps. It reached exact fidelity up to the fifth decimal place while reducing the measurement budget by a factor of 10.
• Transverse-Field Ising Model (6 qubits): In the challenging phase transition regime, MT-QITE improved fidelity by more than two orders of magnitude. The measurement budget was significantly reduced, particularly when leveraging site inversion symmetry with a three-term partition.
• Fermionic Hubbard Model: For various repulsion strengths ($U$), MT-QITE consistently outperformed QITE in both fidelity and measurement count after 10 steps, demonstrating robustness even in regimes with strong electron correlations.
• H4 Chain (Quantum Chemistry): Using a UCCGSD ansatz on 8 qubits, MT-QITE with a three-term partition achieved chemical accuracy across a range of interatomic distances (0.60–1.20 Å), including stretched geometries where standard QITE and unpartitioned MT-QITE failed to converge. Notably, partitioning a Hamiltonian with non-local interactions provided a computational advantage.

Significance and Claims
The paper claims that MT-QITE substantially improves both the fidelity of the resulting ground state and the measurement overhead compared to previously published QITE algorithms, while preserving the method's deterministic character and independence from ad hoc ansatzes.

Key contributions highlighted by the authors include:

• Deterministic Improvement: MT-QITE offers a deterministic path to higher fidelity without the probabilistic overhead of methods like Probabilistic ITE (PITE) or the exponential gate depth of Double-Bracket QITE.
• Parallelizability: The fixed reference state strategy enables the parallel execution of RTE parameter determination for different Hamiltonian terms, a feature not present in standard QITE.
• Efficiency in Non-Local Systems: Contrary to intuition, the authors demonstrate that partitioning Hamiltonians with non-local interactions (as seen in the H4 chain) can yield computational advantages by reducing the linear system size and leveraging the multiple-time variational freedom.
• Applicability: The algorithm is suited for both Noisy Intermediate-Scale Quantum (NISQ) devices and future fault-tolerant quantum computers, offering a route to reduce latency and measurement costs in digital quantum simulation.

The authors conclude that MT-QITE represents an ideal candidate for scenarios requiring high-fidelity ground state preparation at constant circuit depth, avoiding the approximations and probabilistic subroutines found in other approaches.