Quantum Algorithm Achieves 0.4% Accuracy with Tensor-Based Phase Difference Estimation

Scientists are continually seeking more efficient methods for phase estimation, a crucial process in quantum computing and time series analysis. Shu Kanno, Kenji Sugisaki, and Rei Sakuma, alongside et al. from institutions including Mitsubishi Chemical Corporation, Deloitte Tohmatsu LLC, and Keio University, present a novel algorithm leveraging tensor-network circuit compression to improve both the scalability and accuracy of phase estimation, specifically, algorithms of the Quantum Phase Estimation (QPE) type. Their research details a technique utilising time-evolution data extracted through nearest-neighbor gate circuits, enhanced by algorithmic error mitigation and iterative circuit optimisation, demonstrating impressive accuracy improvements of 0.4, 4.7% in simulations of the 8-qubit Hubbard model. Significantly, the team successfully implemented this algorithm on IBM’s Heron devices with up to 52 qubits, representing a major step forward for near-term quantum computing and paving the way for future error-corrected systems.

This breakthrough centres on a technique called quantum phase-difference estimation (QPDE), focusing on energy gap calculations without the need for complex controlled operations for time evolution, a key simplification for current quantum hardware. The study establishes a pathway towards practical applications and prepares for the future era of error-corrected devices by efficiently compressing circuits for initial state preparation and time evolution using matrix product states (MPS) and matrix product operators (MPO).

The core of this work lies in combining tensor networks with QPDE, enabling the largest demonstration of a QPE-type algorithm to date, scaling up to 52 qubits. Researchers implemented algorithmic error mitigation techniques to improve the accuracy of time-evolution and state-preparation circuits, addressing the inherent challenges of approximation errors introduced by tensor compression. Furthermore, an iterative circuit optimization method, combined with merging into matrix product states, was developed to enhance the overlap and efficiency of state preparation, effectively managing the exponential growth of classical computational cost. Verifications performed using a noiseless simulator for an 8-qubit one-dimensional Hubbard model, incorporating an ancilla qubit, revealed accuracies ranging from 0.4, 4.7% error from the true energy gap, demonstrating the effectiveness of the proposed error mitigation strategies.
Experiments conducted on IBM Heron devices, utilising Q-CTRL error suppression, successfully demonstrated the algorithm’s performance on 8-, 36-, and 52-qubit models, employing over 5,000 two-qubit gates. This achievement represents a substantial leap forward, exceeding the limitations of classical full configuration interaction (FCI) methods, which are restricted to a maximum of 46 qubits. The team employed direct data fitting as the classical signal processing method, extracting signals indicative of energy gap information at various time steps. To further refine the accuracy of both time-evolution and state-preparation circuits, they implemented techniques rooted in algorithmic error mitigation and iterative circuit optimization, ultimately merging results into matrix product states. Experiments began with the construction of circuits applying a state preparation operator, Uprep |0⟩⊗N+1, followed by a phase gate P(θ) and the time evolution operator Uevol, concluding with measurement of all zeros.

Uprep and Uevol were meticulously designed using brick-wall gates with depths dprep and devol, respectively. The team extracted a time-series signal, st, containing energy gap information from the compressed circuit, then calculated the energy gap using classical signal processing, specifically, data denoted as {(t, st)}tmax t=t1, where t1 represents ∆t and tmax signifies a maximum time. Verification using a noiseless simulator for an 8-qubit one-dimensional Hubbard model, incorporating an ancilla qubit, demonstrated the algorithm’s ability to achieve accuracies with 0.4, 4.7% error from the true energy gap, contingent on an appropriate time-step size. Notably, the study revealed accuracy improvements attributable to the implemented algorithmic error mitigation strategies.

The researchers also confirmed enhancement of overlap with matrix product states through iterative optimization, demonstrating the effectiveness of their circuit refinement process. Furthermore, the proposed algorithm was successfully demonstrated on Heron devices with Q-CTRL error suppression for 8-, 36-, and 52-qubit models, utilising over 5,000 two-qubit gates. The team calculated the expectation values of the circuits as a function of θ, expressed as m(θ), and combined these values at θ = 0, π/2, π, and 3π/2 to generate the time signal data, st, using the equation st = ⟨φ|(U†evol)t∆t|φ⟩⟨φ’|(Uevol)t∆t|φ’⟩. To account for depolarizing noise, the state was transformed using τ’(t) = (1 −pdep(t))τ(t) + pdep 2N+1 I⊗N+1, where pdep represents the error rate, and the measured probability, m’(θ), was adjusted accordingly. The energy gap, ∆J, was then extracted from the time series data st using the matrix pencil method, a classical signal processing technique, enabling precise determination of energy levels within the quantum system. These large-scale demonstrations for a QPE-type algorithm signify substantial progress towards practical near-term quantum computing and preparation for the era of error-corrected devices.

Tensor-network compression boosts quantum phase estimation accuracy significantly

Scientists have developed a novel phase-difference estimation algorithm based on tensor-network circuit compression, achieving significant advancements in scalability and accuracy for phase estimation (QPE)-type algorithms. The research team constructed circuits using only nearest-neighbor gates and extracted time-evolution data through four-type circuit measurements, paving the way for more efficient quantum computations. To further refine the accuracy of both time-evolution and state-preparation circuits, they introduced techniques leveraging algorithmic error mitigation and iterative circuit optimization combined with merging into matrix product states. Verifications performed using a noiseless simulator on an 8-qubit one-dimensional Hubbard model, incorporating an ancilla qubit, demonstrated accuracies with errors ranging from 0.4% to 4.7% from the true energy gap, dependent on the chosen time-step size.

These results clearly show the effectiveness of the proposed algorithm in accurately determining energy gaps, a crucial step in many quantum simulations. The team observed measurable accuracy improvements directly attributable to the implemented algorithmic error mitigation techniques, confirming their positive impact on the overall performance. Furthermore, iterative optimization demonstrably enhanced the overlap with matrix product states, indicating improved state preparation fidelity. Experiments on Heron devices, utilising Q-CTRL error suppression, successfully demonstrated the algorithm’s performance on 8-, 36-, and 52-qubit models, employing over 5,000 two-qubit gates.

Noiseless simulations with a time-step of 0.05 achieved absolute relative errors of 0.012 at 50 steps and 0.001 at 100 steps, corresponding to error ratios of 4.7% and 0.4% respectively, targeting an accuracy of T/100 = 0.01. Analysis of slice and sweep dependencies revealed that the algorithm maintained stable fluctuations around step 50, except for a time-step of 0.5 where instability arose after 60 steps due to unitary-approximation errors. The team evaluated overlap enhancement for a 37-qubit circuit, achieving a final overlap of 0.62 with a depth of dprep = 24, and further increasing this to 0.81 with dprep = 12 and 1000 sweeps, demonstrating the potential for preparing states with significantly improved overlap compared to previous methods. Real device demonstrations on the 9-qubit circuit with a time-step of 0.1 yielded accuracies on the order of 10⁻³ with error mitigation, compared to 0.033 without it, using 1699 and 1698 two-qubit gates respectively. For the 37- and 53-qubit circuits, the error remained below 0. To further refine the accuracy of both time-evolution and state-preparation circuits, researchers introduced techniques incorporating algorithmic error mitigation and iterative circuit optimisation combined with matrix product state merging. Numerical validations, performed using a noiseless simulator on an 8-qubit one-dimensional Hubbard model with an ancilla qubit, demonstrated accuracies ranging from 0.4, 4.7% error from the true energy gap with appropriate time-step selection.

The algorithm was also successfully demonstrated on Heron devices with Q-CTRL error suppression for 8-, 36-, and 52-qubit models, utilising over 5,000 two-qubit gates, representing a significant advancement for large-scale QPE-type algorithms and preparation for future error-corrected devices. The authors acknowledge limitations including substantial classical computational overhead associated with algorithmic error mitigation and state-preparation accuracy improvements. Future research will focus on achieving higher accuracy and scaling implementations through more efficient algorithms on both quantum and classical fronts, potentially exploring alternative tensor networks and reducing sampling costs.