Quantum amplitude estimation is one of the core subroutines in quantum algorithms. This paper gives a parallelized amplitude estimation (PAE) algorithm that simultaneously achieves near-Heisenberg scaling in the total number of queries and sub-linear scaling in the circuit depth, with respect to the estimation precision. The algorithm is composed of a global GHZ state followed by separated low-depth Grover circuits optimized by quantum signal processing techniques; the number of qubits in the GHZ state and the depth of each circuit is tunable as a trade-off way, which particularly enables even near-Heisenberg-limited and logarithmic-depth algorithm for amplitude estimation. We prove that this trade-off scaling is nearly optimal with use of the parallel quantum adversary method, against folklore on the impossibility of efficient parallelization in amplitude estimation. The proposed algorithm has a form of distributed quantum computing, which may be suitable for device implementation.
Quantum amplitude estimation is one of the core subroutines in quantum algorithms. This paper gives a parallelized amplitude estimation (PAE) algorithm that simultaneously achieves near-Heisenberg scaling in the total number of queries and sub-linear scaling in the circuit depth, with respect to the estimation precision. The algorithm is composed of a global GHZ state followed by separated low-depth Grover circuits optimized by quantum signal processing techniques; the number of qubits in the GHZ state and the depth of each circuit is tunable as a trade-off way, which particularly enables even near-Heisenberg-limited and logarithmic-depth algorithm for amplitude estimation. We prove that this trade-off scaling is nearly optimal with use of the parallel quantum adversary method, against folklore on the impossibility of efficient parallelization in amplitude estimation. The proposed algorithm has a form of distributed quantum computing, which may be suitable for device implementation.
We develop a class of amplitude estimation algorithms with tunable circuit depth M and circuit repetitions N, requiring neither ancilla qubits nor controlled Grover operations. For additive error ε in the Grover angle, they attain the nearly optimal query-depth tradeoff M2N∈O(ε−2) uniformly over λ∈[0,π/2], spanning the full range from classical sampling at M=1 to the Heisenberg limit at M=Θ(ε−1). While prior depth-tunable work establishes comparable angle-accuracy guarantees only away from the boundaries λ→0 or π/2 or at discrete depth tradeoffs, our guarantee extends to both boundaries, so the quantum speedup persists there rather than degrading to classical sampling. Numerical experiments confirm the predicted uniform angle accuracy and show low overhead in practice, making them strong candidates for practical amplitude estimation in the early fault-tolerant regime, with applications such as overlap certification, trial-state verification, and Monte Carlo methods.
Amplitude estimation, in its original form, is formulated as phase estimation upon the Grover iterate. Subsequent improvements to the algorithm have eliminated the need for phase estimation and introduced low-depth variants that trade speedups for lower circuit depth. We make the key observation that amplitude estimation is equivalent to estimating the energy gap of an effective Hamiltonian, whereby discrete-time evolution is generated by amplitude amplification. This enables us to develop an amplitude estimation algorithm for both Heisenberg-limited and low-depth circuit regimes, inspired by statistical phase estimation techniques developed for early fault-tolerant ground-state energy estimation. In the Heisenberg-limited regime, our approach achieves performance comparable to state-of-the-art methods while using simplified classical post-processing. In the low-depth regime, it obtains optimal query--depth tradeoffs up to polylogarithmic factors, with provable guarantees and improved empirical performance over prior approaches. The resulting protocol is ancilla-free and requires only standard Grover reflections. Due to its flexibility, generality, and robustness, we expect our approach to be a key enabler for a broad range of early fault-tolerant applications.
Standard quantum amplitude estimation algorithms provide quadratic speedup to Monte-Carlo simulations but require a circuit depth that scales as inverse of the estimation error. In view of the shallow depth in near-term devices, the precision achieved by these algorithms would be low. In this article, we bypass this limitation by performing the classical Monte-Carlo method on the quantum algorithm itself, achieving a higher than classical precision using low-depth circuits. We require the quantum algorithm to be weakly biased in order to avoid error accumulation during this process. Our method is parallel and can be as weakly biased as the constituent algorithm in some cases.
Amplitude Estimation (AE) is a critical subroutine in many quantum algorithms, allowing for a quadratic speedup in various applications like those involving estimating statistics of various functions as in financial Monte Carlo simulations. Much work has gone into devising methods to efficiently estimate the amplitude of a quantum state without expensive operations like the Quantum Fourier Transform (QFT), which is especially prohibitive given the constraints of current NISQ devices. Newer methods have reduced the number of operations required on a quantum computer and are the most promising near-term implementations of the AE subroutine. While it remains to be seen the exact circuit requirements for a quantum advantage in applications relying on AE, it is necessary to continue to benchmark the algorithm's performance on current quantum computers and the circuit costs associated with such subroutines. Given these considerations, we expand on results from previous experiments in using Maximum Likelihood Estimation (MLE) to approximate the amplitude of a quantum state and provide empirical upper bounds on the current feasible circuit depths for AE on a superconducting quantum computer. Our results show that MLE using optimally-compiled circuits can currently outperform naive sampling for up to 3 Grover Iterations with a circuit depth of 131, which is higher than reported in other experimental results. This functional benchmark is one of many that will be continually monitored against current quantum hardware to measure the necessary progress towards quantum advantage.
Non-adaptive quantum amplitude estimation (QAE) fixes its Grover depths in advance, so every circuit can run in parallel, but it has so far needed more queries than the best adaptive methods. We show that most of this gap comes from two conventional choices: subspace-based post-processing and power-of-two depth ladders. We replace the first by the exact maximum-likelihood estimate, one matrix multiplication per batch of estimates, and the second by a geometric ladder with ratio r≈1.45. The result is a fully parallel, deterministic-schedule estimator with total query complexity 2.8-3.1/ε at 95% confidence for target errors from 3.5×10−3 to 10−6. This matches the average-case complexity of chebAE, the best benchmarked adaptive method, within statistical uncertainty (with the lower point estimate at every scale tested), beats its maximum-observed complexity by 1.6×, and needs a maximum sequential depth of only 0.21/ε against chebAE's 2.9/ε. Relative to csAE, the best non-adaptive benchmark, the constants improve by 30-35% at 95% and 1.5-1.7× at 99% confidence. The optimal ratio has a simple origin. Doubling is the fastest depth growth at which the data can still tell neighboring candidate values apart, so power-of-two ladders sit at the edge of confusion and must buy reliability with extra shots; a slightly denser ladder checks every scale redundantly. An error-probability analysis reproduces the measured failure rates and locates the optimum. The likelihood formulation extends directly to noise-aware estimation, and uniformly scaling the capped ladder covers the depth-limited regime, realizing the optimal trade-off MNtot≈(0.4-0.6)/ε2 within ∼1.1× of the schedule's Cramér-Rao limit.
We demonstrate that the problem of amplitude estimation, a core subroutine used in many quantum algorithms, can be mapped directly to a problem in signal processing called direction of arrival (DOA) estimation. The DOA task is to determine the direction of arrival of an incoming wave with the fewest possible measurements. The connection between amplitude estimation and DOA allows us to make use of the vast amount of signal processing algorithms to post-process the measurements of the Grover iterator at predefined depths. Using an off-the-shelf DOA algorithm called ESPRIT together with a compressed-sensing based sampling approach, we create a phase-estimation free, parallel quantum amplitude estimation (QAE) algorithm with a worst-case sequential query complexity of ∼4.3/ε and a parallel query complexity of ∼0.26/ε at 95% confidence. This performance is statistically equivalent and a 16× improvement over Rall and Fuller [Quantum 7, 937 (2023)], for sequential and parallel query complexity respectively, which to our knowledge is the best published result for amplitude estimation. The approach presented here provides a simple, robust, parallel method to performing QAE, with many possible avenues for improvement borrowing ideas from the wealth of literature in classical signal processing.