Efficient quantum phase estimation with adaptive entanglement-assisted Hadamard test

The adaptive entanglement-assisted Hadamard test (AEHT) is proposed as an efficient method for high-precision quantum phase estimation that overcomes the limitations of conventional fixed-amplification entanglement-assisted Hadamard tests by iteratively refining the reference phase to enable progressively stronger amplification, leading to a device-restart count scaling as O(1/ϵ) instead of O(1/ϵ 2) in the hardware-unconstrained regime.

Introduction and Motivation

Quantum phase estimation (QPE) is fundamental for many quantum algorithms, but conventional QPE demands coherence times that scale exponentially with desired precision and are highly sensitive to experimental noise. Standard Hadamard test (SHT) provides a practical alternative, requiring O(1/ϵ 2) shot counts to estimate a phase to accuracy ϵ. Entanglement-assisted Hadamard test (EHT) uses an m-qubit Greenberger-Horne-Zeilinger (GHZ) entangled state to reduce the required shot counts to O(1/(mϵ) 2). However, the feasibility of large GHZ states is limited because a coarse reference phase permits only a small amplification, preventing the direct use of large GHZ states even when sufficient hardware resources are available. The AEHT addresses this by iteratively refining the reference phase, enabling progressively stronger amplification as the iteration goes by.

Adaptive Entanglement-Assisted Hadamard Test (AEHT)

The AEHT starts from a coarse reference phase and uses a modestly sized GHZ state whose amplification is safe given the current uncertainty. The measurements sharpen the estimate, and this improved estimate becomes the reference for the next round. This process allows for progressively larger amplification as the reference improves, keeping required shot counts under control even at high precision. The algorithm is structured iteratively over T rounds to achieve a total accuracy of ϵ with a success probability of at least 1 − pfail.

Error Analysis and Bias Suppression

When the prepared state is imperfect, the measured overlap takes the form ρeiθψ ψ, where ρ = ⟨ψUψ⟩ is the amplitude. This introduces an amplitude-induced bias εbias which persists even if shot counts increase, unlike statistical error εstat which can be controlled. The AEHT separates the irreducible preparation error from the reducible algorithmic error by iteratively updating the reference phase and jointly optimizing amplification level and shot count. The protocol ensures that the amplitude-induced algorithmic bias is forced to remain within its prescribed round-wise bias budget, which is managed by choosing an amplification mt based on binary search over feasible values.

Device-Restart Count Comparison

The cost of quantum phase estimation is measured by the device-restart count rather than shot count, as this accounts for parallel execution capacity on a finite qubit budget. For fixed-m EHT, the restart count grows as O(1/ϵ 2). In contrast, the AEHT scheme keeps the perround shot budget roughly constant, leading to a total restart count that grows only as O(1/ϵ), up to doubly logarithmic factors. This demonstrates that the AEHT method achieves a superlinear reduction in device-restart count compared with fixed-m EHT in the high-precision regime.

Hardware Constraints and Robustness

The choice of amplification mt is constrained by both the reference error bound ∆t and the hardware capability, where mt = min mideal t, mhw. When saturation occurs at the hardware limit mhw, both schemes operate with a fixed bounded amplification, but AEHT still maintains a constant-factor advantage over fixed-m EHT in this regime. Numerical experiments confirm the effectiveness of the proposed AEHT compared with conventional methods under this measure, showing its robustness under representative circuit noise. The actual AEHT error follows its upper bound and produces an increasingly accurate phase estimate as the device-restart count increases, unlike conventional methods which exhibit a nonvanishing bias floor.

Conclusion

The proposed AEHT provides a hardware-aware route to precision phase estimation with substantially reduced devicerestart overhead on near-term quantum processors. The method unlocks the amplification power of large entangled states, and the maximum amplification of the AEHT method increases from m = 11 at target accuracy ϵ = 10−2 to m = 1250 at target accuracy ϵ = 10−4. The AEHT successfully separates the irreducible preparation error from the statistical error and amplitude-induced bias, which is key to its superior performance. The proposed method provides a hardware-aware route to precision phase estimation with substantially reduced devicerestart overhead on near-term quantum processors.

1. Bold header: Adaptive Phase Estimation for High-Precision QPE

The AEHT algorithm iteratively refines the reference phase, enabling progressively stronger amplification as the iteration goes by, which allows near-term quantum processors to estimate high-precision quantum phases more efficiently than conventional fixed-amplification EHT.

2. Bold header: Bias Suppression in Imperfect State Preparation

By implementing the AEHT, the method suppresses this bias arising from imperfect eigenstate preparation, ensuring that the estimation error continues to decrease round by round even when the amplitude is unknown and not exactly one.

3. Bold header: Hardware-Aware Cost Metric for Quantum Algorithms

The transition from shot count to device-restart count quantifies realization cost better, showing that AEHT yields a restart count that grows only as O(1/ϵ), up to doubly logarithmic corrections in the hardware unconstrained regime, offering a constant-factor advantage over a fixed-amplification method.

4. Bold header: Robust Phase Estimation under Circuit Noise

The adaptive protocol demonstrates robustness against circuit noise by ensuring the amplitude-induced algorithmic bias is forced to remain within the prescribed round-wise bias budget, preventing an error floor that plagues conventional fixed-amplification tests.

5. Bold header: Optimal Amplification Selection Strategy

The paper introduces Algorithm 4 (MINRESTARTAMP) to select the amplification factor by minimizing packed restarts, choosing the feasible amplification with the smallest packed restart count while respecting both branch and hardware constraints.

The entanglement-assisted Hadamard test (EHT) is a practical method for estimating a quantum phase by amplifying the phase signal. However, the feasible amplification is fundamentally limited by the accuracy of the reference phase, such that the method is inefficient in the high-precision regime. In this work, we propose an algorithm, called adaptive entanglement-assisted Hadamard test (AEHT), that iteratively refines the reference phase, enabling progressively stronger amplification as the iteration goes by. We further consider the imperfect eigenstate preparation scenario, where a systematic bias is unavoidable when estimating the quantum phase with the conventional EHT. Such a bias can be suppressed by the proposed AEHT. Moreover, taking physical implementation into consideration, we adopt device-restart count to measure the cost of quantum phase estimation, rather than shot count. The numerical experiments confirm the effectiveness of the proposed AEHT compared with conventional methods under this measure. By unlocking the full amplification power of large entangled states, this work offers an efficient method to estimate high-precision quantum phase on near-term quantum processors.