arXiv:2206.06392

An iterative quantum-phase-estimation protocol for near-term quantum hardware

Smith, Joseph G., C. H. W. Barnes, David R. M. Arvidsson-Shukur

Published Jun 13, 2022• 0 citations

Abstract

Given $N_{\textrm{tot}}$ applications of a unitary operation with an unknown phase $θ$, a large-scale fault-tolerant quantum system can {reduce} an estimate's {error} scaling from $\mathcal{O} \left[ 1 / \sqrt{N_{\textrm{tot}}} \right]$ to $\mathcal{O} \left[ 1 / {N_{\textrm{tot}}} \right]$. Owing to the limited resources available to near-term quantum devices, entanglement-free protocols have been developed, which achieve a $\mathcal{O} \left[ \log(N_{\textrm{tot}}) / N_{\textrm{tot}} \right]$ {mean-absolute-error} scaling. Here, we propose a new two-step protocol for near-term phase estimation, with an improved {error} scaling. Our protocol's first step produces several low-{standard-deviation} estimates of $θ$, within $θ$'s parameter range. The second step iteratively hones in on one of these estimates. Our protocol's {mean absolute error} scales as $\mathcal{O} \left[ \sqrt{\log (\log N_{\textrm{tot}})} / N_{\textrm{tot}} \right]$. Furthermore, we demonstrate a reduction in the constant scaling factor and the required circuit depths: our protocol can outperform the asymptotically optimal quantum-phase estimation algorithm for realistic values of $N_{\textrm{tot}}$.

DOI: 10.48550/arxiv.2206.06392

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arXiv:2206.06392

An iterative quantum-phase-estimation protocol for near-term quantum hardware

Smith, Joseph G., C. H. W. Barnes, David R. M. Arvidsson-Shukur

Published Jun 13, 2022• 0 citations

arXiv HTML PDF

Abstract

Given

N_{\textrm{tot}}

applications of a unitary operation with an unknown phase

θ

, a large-scale fault-tolerant quantum system can {reduce} an estimate's {error} scaling from

\mathcal{O} \left[ 1 / \sqrt{N_{\textrm{tot}}} \right]

\mathcal{O} \left[ 1 / {N_{\textrm{tot}}} \right]

. Owing to the limited resources available to near-term quantum devices, entanglement-free protocols have been developed, which achieve a

\mathcal{O} \left[ \log(N_{\textrm{tot}}) / N_{\textrm{tot}} \right]

{mean-absolute-error} scaling. Here, we propose a new two-step protocol for near-term phase estimation, with an improved {error} scaling. Our protocol's first step produces several low-{standard-deviation} estimates of

θ

, within

θ

's parameter range. The second step iteratively hones in on one of these estimates. Our protocol's {mean absolute error} scales as

\mathcal{O} \left[ \sqrt{\log (\log N_{\textrm{tot}})} / N_{\textrm{tot}} \right]

. Furthermore, we demonstrate a reduction in the constant scaling factor and the required circuit depths: our protocol can outperform the asymptotically optimal quantum-phase estimation algorithm for realistic values of

N_{\textrm{tot}}

DOI: 10.48550/arxiv.2206.06392

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Given ${N}_{\text{tot}}$ applications of a unitary operation, parametrized by an unknown phase, a phase-estimation protocol on a large-scale fault-tolerant quantum system can reduce the standard deviation of an estimate of the phase from scaling as $O[1/\sqrt{{N}_{\text{tot}}}]$ to scaling as $O[1/{N}_{\text{tot}}]$ . Owing to the limited resources available to near-term quantum devices, protocols that do not entangle probes have been developed. Their mean absolute error scales as $O[log({N}_{\text{tot}})/{N}_{\text{tot}}]$ . Here, we propose a two-step protocol for near-term phase estimation, with an improved error scaling. Our protocol's first step produces several low-standard-deviation estimates of $\ensuremath{\theta}$, within $\ensuremath{\theta}$'s parameter range. The second step iteratively homes in on one of these estimates. Our protocol achieves a mean-absolute-error scaling of $O[\sqrt{log(log{N}_{\text{tot}})}/{N}_{\text{tot}}]$ and a root-mean-square-error scaling of $O[\sqrt{log{N}_{\text{tot}}}/{N}_{\text{tot}}]$ . Furthermore, we demonstrate a reduction in the constant scaling factor and the required circuit depths. This allows our protocol to outperform the asymptotically optimal quantum-phase-estimation algorithm for realistic values of ${N}_{\text{tot}}$ .Given ${N}_{\text{tot}}$ applications of a unitary operation, parametrized by an unknown phase, a phase-estimation protocol on a large-scale fault-tolerant quantum system can reduce the standard deviation of an estimate of the phase from scaling as $O[1/\sqrt{{N}_{\text{tot}}}]$ to scaling as $O[1/{N}_{\text{tot}}]$. Owing to the limited resources available to near-term quantum devices, protocols that do not entangle probes have been developed. Their mean absolute error scales as $O[log({N}_{\text{tot}})/{N}_{\text{tot}}]$. Here, we propose a two-step protocol for near-term phase estimation, with an improved error scaling. Our protocol's first step produces several low-standard-deviation estimates of $\ensuremath{\theta}$, within $\ensuremath{\theta}$'s parameter range. The second step iteratively homes in on one of these estimates. Our protocol achieves a mean-absolute-error scaling of $O[\sqrt{log(log{N}_{\text{tot}})}/{N}_{\text{tot}}]$ and a root-mean-square-error scaling of $O[\sqrt{log{N}_{\text{tot}}}/{N}_{\text{tot}}]$. Furthermore, we demonstrate a reduction in the constant scaling factor and the required circuit depths. This allows our protocol to outperform the asymptotically optimal quantum-phase-estimation algorithm for realistic values of ${N}_{\text{tot}}$.

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Joseph G. Smith, C. H. W. Barnes, David R. M. Arvidsson-Shukur

Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary N tot times, the variance of the estimates can scale as O ( 1 / N tot 2 ) , compared to the best “classical” strategy with O ( 1 / N tot ) . The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without interprobe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O ( 1 / N tot 2 ) . Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analizes the results of previous steps. Published by the American Physical Society 2024Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary N tot times, the variance of the estimates can scale as O ( 1 / N tot 2 ) , compared to the best “classical” strategy with O ( 1 / N tot ) . The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without interprobe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O ( 1 / N tot 2 ) . Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analizes the results of previous steps. Published by the American Physical Society 2024

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Given

N_{\textrm{tot}}

applications of a unitary operation with an unknown phase

θ

, a large-scale fault-tolerant quantum system can {reduce} an estimate's {error} scaling from

\mathcal{O} \left[ 1 / \sqrt{N_{\textrm{tot}}} \right]

\mathcal{O} \left[ 1 / {N_{\textrm{tot}}} \right]

. Owing to the limited resources available to near-term quantum devices, entanglement-free protocols have been developed, which achieve a

\mathcal{O} \left[ \log(N_{\textrm{tot}}) / N_{\textrm{tot}} \right]

θ

, within

θ

's parameter range. The second step iteratively hones in on one of these estimates. Our protocol's {mean absolute error} scales as

\mathcal{O} \left[ \sqrt{\log (\log N_{\textrm{tot}})} / N_{\textrm{tot}} \right]

N_{\textrm{tot}}

Given

{N}_{\text{tot}}

applications of a unitary operation, parametrized by an unknown phase, a phase-estimation protocol on a large-scale fault-tolerant quantum system can reduce the standard deviation of an estimate of the phase from scaling as

O[1/\sqrt{{N}_{\text{tot}}}]

to scaling as

O[1/{N}_{\text{tot}}]

. Owing to the limited resources available to near-term quantum devices, protocols that do not entangle probes have been developed. Their mean absolute error scales as

O[log({N}_{\text{tot}})/{N}_{\text{tot}}]

. Here, we propose a two-step protocol for near-term phase estimation, with an improved error scaling. Our protocol's first step produces several low-standard-deviation estimates of $\ensuremath{\theta}$, within $\ensuremath{\theta}$'s parameter range. The second step iteratively homes in on one of these estimates. Our protocol achieves a mean-absolute-error scaling of

O[\sqrt{log(log{N}_{\text{tot}})}/{N}_{\text{tot}}]

and a root-mean-square-error scaling of

O[\sqrt{log{N}_{\text{tot}}}/{N}_{\text{tot}}]

. Furthermore, we demonstrate a reduction in the constant scaling factor and the required circuit depths. This allows our protocol to outperform the asymptotically optimal quantum-phase-estimation algorithm for realistic values of

{N}_{\text{tot}}

N_{\mathrm{tot}}

times, the variance of the estimates can scale as

O(1/{N^2_{\mathrm{tot}}})

, compared to the best ``classical'' strategy with

O(1/{N_{\mathrm{tot}}})

. The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without inter-probe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling

O(1/{N^2_{\mathrm{tot}}})

. Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analyses the results of previous steps.

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\le k \le L\}

of shots per bit is:

\arg \max_{\{N_k\}} \mathbb{P}(\text{all } L \text{ bits correct}) \text{ s.t.} \sum_{k=1}^L w_k N_k \le 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\ll 1

. We observe that when

W\gg \sum_k w_k \ln\! \left(\sum_k w_k\right)

, the optimal

N_k^*

is proportional to

\frac{\ln\! \left(\sum_k w_k\right)}{c_k}

and does not depend on individual

w_k

, where

c_k=-\ln\left(2\sqrt{\left(1-\frac{q_k}{2} \right) \frac{q_k}{2}}\right) \propto \ln\frac{1}{q_k}

. However, when

\sum_k w_k \ll W\ll\sum_k w_k \ln\! \left(\sum_k w_k\right)

, the allocation is also (additively) influenced by

\frac{1}{c_k}\ln\frac{1}{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.

t

scaling for circuit depth

\mathcal{O}(t \text{ polylog}(t N/ε))

, while reducing the control overhead from a multiplicative to an additive factor. We report that this compression protocol enables the implementation of Iterative Quantum Phase Estimation with as few as 414 CNOT gates for a frustrated quantum spin system on a 6$\times$6 triangular lattice and delivers ground state energy errors below 1% (with

\pm

1.5% variation, calculated with a hardware noise aware pipeline) on a 4$\times$4 triangular lattice using the noisy emulator of the Quantinuum H2 trapped ion device.