⚛️ quantum physics

Efficient quantum phase estimation with adaptive entanglement-assisted Hadamard test

This paper proposes the Adaptive Entanglement-Assisted Hadamard Test (AEHT), an algorithm that iteratively refines the reference phase to overcome the precision limitations and systematic biases of conventional methods, thereby enabling efficient high-precision quantum phase estimation on near-term devices using device-restart counts as a cost metric.

Original authors: Hengzhun Chen, Benchi Zhao, Yingzhou Li

Published 2026-10-02

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CC BY 4.0

Original authors: Hengzhun Chen, Benchi Zhao, Yingzhou Li

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Quantum computers promise to solve problems that would take classical machines thousands of years, but they are notoriously fragile. To work, they must hold onto delicate states of information long enough to perform calculations, a requirement that often clashes with the noisy reality of current hardware. One of the most fundamental tasks for these machines is quantum phase estimation, a process used to extract specific values from quantum systems. This calculation is the engine behind famous algorithms for breaking codes and simulating molecules, yet the standard way of doing it requires the computer to maintain perfect coherence for an impossibly long time. Because today's machines cannot sustain such long periods of stability, scientists have turned to alternative methods that use shorter, more manageable circuits, even if those methods require repeating the experiment many times to get a precise answer.

The challenge with these shorter methods is a trade-off between speed and accuracy. Researchers can speed up the process by using a technique that links multiple particles together in a special state, effectively amplifying the signal they are trying to measure. However, this amplification is dangerous if the scientists do not have a very good guess to start with. If the initial guess is too far off, the amplified signal becomes confusing and impossible to interpret, forcing the researchers to use a much smaller, slower amplification. This creates a bottleneck: to get a highly precise result, one needs a large amplification, but one cannot use a large amplification without a very precise starting guess, which is exactly what the measurement is supposed to provide.

In a new study, researchers Hengzhun Chen, Benchi Zhao, and Yingzhou Li have developed a solution to this circular problem called the adaptive entanglement-assisted Hadamard test. Instead of sticking with a single, fixed level of amplification, their method acts like a refining loop. The process begins with a rough guess and a small, safe amount of amplification. The computer performs the measurement, which sharpens the estimate of the true value. This improved estimate then becomes the new starting point for the next round, allowing the researchers to safely use a larger amplification factor. By repeating this cycle, the method progressively increases the amplification power as the estimate becomes more accurate, unlocking the ability to use large groups of entangled particles without the risk of confusion.

The team also addressed a common real-world issue where the quantum state prepared by the machine is not perfect. In such cases, the signal is often weaker than expected, which introduces a systematic error that cannot be fixed simply by running the experiment more times. Their adaptive approach successfully suppresses this bias, allowing the error to decrease with every round of the process. To measure the true cost of these experiments, the authors introduced a new metric called the device-restart count. On modern quantum processors, the most time-consuming part of an experiment is not the number of times a circuit is run, but the number of times the entire machine must be reset and reloaded. Their simulations show that by using their adaptive method, the number of these costly resets drops significantly compared to traditional fixed methods, especially when high precision is required.

The researchers tested their idea through numerical simulations on a virtual quantum processor with 2,500 qubits. They found that for exact quantum states, their method reduced the number of required machine resets by a factor that grows as the desired precision increases, offering a massive efficiency gain over standard techniques. Even when the quantum states were imperfect, the adaptive method continued to improve the accuracy, whereas traditional methods hit a hard limit where no amount of extra data could remove the error. The study concludes that by adapting the reference phase to the current estimate and measuring cost in terms of machine resets rather than just raw data points, this approach offers a practical path to high-precision quantum calculations on the noisy, imperfect hardware available today.

Technical Summary: Efficient Quantum Phase Estimation with Adaptive Entanglement-Assisted Hadamard Test

Problem Statement
Quantum Phase Estimation (QPE) is a fundamental subroutine for algorithms such as Shor's factoring and quantum simulation. However, conventional QPE requires deep circuits and coherence times that scale exponentially with desired precision, rendering it impractical for near-term quantum devices. The Standard Hadamard Test (SHT) offers a shallow-circuit alternative but suffers from a shot count complexity of $O(1/\epsilon^2)$ to achieve accuracy $\epsilon$ . The Entanglement-Assisted Hadamard Test (EHT) improves this by using an $m$ -qubit Greenberger-Horne-Zeilinger (GHZ) state to amplify the phase signal, theoretically reducing the shot count to $O(1/(m\epsilon)^2)$ .

Despite this advantage, the EHT faces a critical limitation: the feasible amplification factor $m$ is strictly bounded by the accuracy of the reference phase $\vartheta$ . Specifically, to avoid ambiguity in the $\arcsin$ function, the condition $m|\theta - \vartheta| < \pi/2$ must hold. In high-precision regimes, a coarse reference phase forces $m$ to be small, negating the benefits of entanglement. Furthermore, in realistic scenarios where the prepared state is not an exact eigenstate, the EHT signal is attenuated by an unknown amplitude $\rho < 1$ . This introduces a systematic bias that cannot be eliminated by increasing the shot count, creating an error floor for conventional methods. Finally, existing cost metrics based on "shot count" fail to account for the parallel execution capabilities of large quantum processors, where the true bottleneck is often the "device-restart count" (the number of times the hardware must be initialized and reset).

Methodology
The authors propose the Adaptive Entanglement-Assisted Hadamard Test (AEHT), an iterative algorithm designed to overcome the reference-phase limitation and handle imperfect state preparation.

Iterative Reference Refinement: The algorithm begins with a coarse reference phase $\vartheta_0$ and a known uncertainty bound $\Delta_0$ . In each round $t$ , it selects an amplification factor $m_t$ that is safe given the current uncertainty (i.e., $m_t \Delta_t \leq \chi$ , where $\chi < \pi/2$ ). The measurements from this round refine the phase estimate, which then becomes the reference $\vartheta_{t+1}$ for the next round. As the reference improves (uncertainty $\Delta_t$ decreases), the algorithm permits a larger $m_t$ , enabling progressively stronger amplification.
Handling Imperfect States: For scenarios where the prepared state $|\psi\rangle$ is not an exact eigenstate, the signal is attenuated by an amplitude $\rho$ . The AEHT decomposes the error into preparation error (irreducible), statistical error, and amplitude-induced bias. By explicitly budgeting for the bias and choosing $m_t$ such that the bias remains within a prescribed fraction of the target accuracy, the algorithm systematically suppresses the bias in each iteration, unlike fixed-amplification methods which hit a bias floor.
Cost Metric: The paper introduces the device-restart count as the primary cost metric. This accounts for the fact that on a processor with $Q$ qubits, multiple independent circuit instances can run in parallel. The number of restarts $R$ is calculated based on the total shot count divided by the parallel capacity $\kappa(m)$ , which depends on the GHZ size $m$ and system qubits.

Key Contributions

Adaptive Amplification: The AEHT algorithm dynamically increases the GHZ state size as the reference phase becomes more precise, unlocking the full amplification power of large entangled states without requiring prior knowledge of the phase.
Bias Suppression: The method provides a theoretical framework to suppress amplitude-induced bias in imperfect state preparation scenarios, allowing the estimation error to decrease continuously with iterations, whereas conventional EHTs exhibit a non-vanishing error floor.
Hardware-Aware Cost Analysis: By shifting the cost metric from shot count to device-restart count, the paper demonstrates that the adaptive scheme offers a significant advantage in parallelizable hardware architectures.
Complexity Bounds:
- For exact eigenstates, the total shot count scales as $O(\log(1/\epsilon))$ (up to doubly logarithmic factors) when hardware limits are not reached.
- In terms of device-restart count, the adaptive scheme scales as $O(1/\epsilon)$ (up to logarithmic factors), compared to the $O(1/\epsilon^2)$ scaling of fixed-amplification EHT.
- When the GHZ size is limited by the available qubits ( $m$ saturates), the scaling reverts to $O(1/\epsilon^2)$ , but the adaptive method retains a constant-factor advantage over fixed-amplification methods.

Results
Numerical experiments were conducted using a simulated quantum processor with $Q=2500$ qubits.

Exact Eigenstates: The AEHT method demonstrated a superlinear reduction in device-restart counts compared to SHT and fixed-m EHT. As the target accuracy $\epsilon$ tightened, the maximum amplification $m$ increased (e.g., from $m=11$ at $\epsilon=10^{-2}$ to $m=1250$ at $\epsilon=10^{-4}$ ), maintaining a restart count advantage of roughly two orders of magnitude even after saturation.
Imperfect Eigenstates: In the presence of state preparation infidelity ( $\eta=0.01$ ), fixed-m EHT and SHT methods hit a bias floor, requiring infinite restarts to surpass it. In contrast, AEHT successfully suppressed the bias, achieving higher accuracy with fewer restarts.
Noise Robustness: Simulations including depolarizing noise and readout errors confirmed that while SHT errors converge to a worse value due to noise accumulation, AEHT errors continue to follow the theoretical upper bound, demonstrating robustness.

Significance
The paper claims to offer an efficient method for estimating high-precision quantum phases on near-term quantum processors. By adapting the reference phase to the current estimate and utilizing device-restart counts as a realistic cost metric, the work makes entanglement-assisted phase estimation practical in the high-precision regime. The authors suggest that this approach is particularly well-suited for platforms like neutral-atom arrays, where parallel execution is possible but initialization and rearrangement are costly. The paper concludes by noting that future work could involve demonstrating the AEHT method on actual quantum hardware.

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