Optimal Resource Scaling for Early Fault Tolerant Iterative Quantum Phase Estimation under Cost Error Tradeoffs
Abstract
The iterative quantum phase estimation algorithm (IPEA) is better suited for NISQ and early fault-tolerant hardware compared to the original algorithm, which requires the inverse quantum Fourier transform. However, current error mitigation and correction implementations lead to imperfect unitaries, causing erroneous phase feedback in IPEA. To improve the reliability of phase bits is to repeat the k^th iteration for the 2^k-th power of the unitary Nk* times, followed by a majority decision. However, due to variations in their circuit complexities, different iterations incur varying resource costs (wk) and different per-shot error probabilities (qk). Given tight resource constraints, we ask what the optimal number {Nk*: 1≤ k ≤ L} of shots per bit is: arg max{Nk} ℙ(all L bits correct) s.t. ∑k=1L wk Nk ≤ W We obtain closed-form expressions for N*k under the conditions: $W \gg \sum_k w_k $(enough for at least one shot per iteration), wk increases with k, qk are small enough and L is large. We do so by analyzing tractable upper and lower surrogate problems derived via tight concentration and anti-concentration bounds and showing that their solutions converge when qk≪ 1. We observe that when W≫ ∑k wk ln (∑k wk), the optimal Nk* is proportional to (ln (∑k wk))/ck and does not depend on individual wk, where ck=−ln(2√((1−qk/2 ) qk/2)) ∝ ln1/qk. However, when ∑k wk ≪ W≪∑k wk ln (∑k wk), the allocation is also (additively) influenced by 1/ckln1/wk for larger values of k, i.e., higher powers of unitary. These results lead to simple rules of thumb for resource allocation that may benefit the design of practical systems.
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BibTeX
@misc{mahindrakar2026optimal,
title = {{Optimal Resource Scaling for Early Fault Tolerant Iterative Quantum Phase Estimation under Cost Error Tradeoffs}},
author = {Mrudula A Mahindrakar and Avhishek Chatterjee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40128},
eprint = {2609.40128},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40128},
}
RIS
TY - UNPB
TI - Optimal Resource Scaling for Early Fault Tolerant Iterative Quantum Phase Estimation under Cost Error Tradeoffs
AU - Mrudula A Mahindrakar
AU - Avhishek Chatterjee
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40128
UR - https://arxiv.org/abs/2609.40128
AB - The iterative quantum phase estimation algorithm (IPEA) is better suited for NISQ and early fault-tolerant hardware compared to the original algorithm, which requires the inverse quantum Fourier transform. However, current error mitigation and correction implementations lead to imperfect unitaries, causing erroneous phase feedback in IPEA. To improve the reliability of phase bits is to repeat the k^th iteration for the 2^k-th power of the unitary N_k^* times, followed by a majority decision. However, due to variations in their circuit complexities, different iterations incur varying resource costs (w_k) and different per-shot error probabilities (q_k). Given tight resource constraints, we ask what the optimal number {N_k^*: 1≤ k ≤ L} of shots per bit is: arg max_{N_k} ℙ(all L bits correct) s.t. ∑_k=1^L w_k N_k ≤ W We obtain closed-form expressions for N^*_k under the conditions: $W \gg \sum_k w_k $(enough for at least one shot per iteration), w_k increases with k, q_k are small enough and L is large. We do so by analyzing tractable upper and lower surrogate problems derived via tight concentration and anti-concentration bounds and showing that their solutions converge when q_k≪ 1. We observe that when W≫ ∑_k w_k ln (∑_k w_k), the optimal N_k^* is proportional to (ln (∑_k w_k))/c_k and does not depend on individual w_k, where c_k=−ln(2√((1−q_k/2 ) q_k/2)) ∝ ln1/q_k. However, when ∑_k w_k ≪ W≪∑_k w_k ln (∑_k w_k), the allocation is also (additively) influenced by 1/c_kln1/w_k for larger values of k, i.e., higher powers of unitary. These results lead to simple rules of thumb for resource allocation that may benefit the design of practical systems.
ER -