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Adaptive quantum phase estimation can be better than non-adaptive

Noah Linden, Ronald de Wolf

TL;DR

The paper investigates adaptive versus non-adaptive quantum phase estimation under a promise on the permissible phase values $\phi\in[0,1)$, showing that adaptivity can reduce the number of uses of the unitary $U_φ$ by about a factor of 2 relative to non-adaptive strategies. It first establishes a concrete 4-phase example where adaptive methods need roughly $\tfrac{1}{2}2^k$ uses while non-adaptive strategies require about $\tfrac{2}{3}2^k$, and then generalizes to phase-sets $F_ℓ$ with $2^ℓ$ pairs, preserving a near-factor-2 gap. The authors provide tight upper and lower bounds for both adaptive and non-adaptive approaches, employing novel duality-based arguments (Farkas’s lemma) to rule out efficient non-adaptive constructions. They also discuss the limits of larger separations, the potential for even more pronounced adaptive advantages in other quantum information tasks, and avenues for future work on noise and scaling beyond the idealized setting.

Abstract

Quantum phase estimation is one of the most important tools in quantum algorithms. It can be made non-adaptive (meaning all applications of the unitary $U_φ$ happen simultaneously) without using more applications of $U_φ$, albeit at the expense of using many more qubits. It is also known that there is no advantage for adaptive algorithms in the case where the phase that needs to be estimated is arbitrary or is uniformly random. Here we give examples of a special case of phase estimation, with a promise on the values that the unknown phase can take, where adaptive methods are provably better than non-adaptive methods by a factor of nearly 2 in the number of uses of $U_φ$. We also prove some upper bounds on the maximum advantage that adaptive algorithms for phase estimation can achieve over non-adaptive ones.

Adaptive quantum phase estimation can be better than non-adaptive

TL;DR

The paper investigates adaptive versus non-adaptive quantum phase estimation under a promise on the permissible phase values , showing that adaptivity can reduce the number of uses of the unitary by about a factor of 2 relative to non-adaptive strategies. It first establishes a concrete 4-phase example where adaptive methods need roughly uses while non-adaptive strategies require about , and then generalizes to phase-sets with pairs, preserving a near-factor-2 gap. The authors provide tight upper and lower bounds for both adaptive and non-adaptive approaches, employing novel duality-based arguments (Farkas’s lemma) to rule out efficient non-adaptive constructions. They also discuss the limits of larger separations, the potential for even more pronounced adaptive advantages in other quantum information tasks, and avenues for future work on noise and scaling beyond the idealized setting.

Abstract

Quantum phase estimation is one of the most important tools in quantum algorithms. It can be made non-adaptive (meaning all applications of the unitary happen simultaneously) without using more applications of , albeit at the expense of using many more qubits. It is also known that there is no advantage for adaptive algorithms in the case where the phase that needs to be estimated is arbitrary or is uniformly random. Here we give examples of a special case of phase estimation, with a promise on the values that the unknown phase can take, where adaptive methods are provably better than non-adaptive methods by a factor of nearly 2 in the number of uses of . We also prove some upper bounds on the maximum advantage that adaptive algorithms for phase estimation can achieve over non-adaptive ones.

Paper Structure

This paper contains 18 sections, 47 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Phase estimation
  3. A basic problem in sensing
  4. Our result: adaptive phase-estimation algorithms can be twice as efficient
  5. Other examples of adaptive advantage
  6. First example: a set $F$ with 4 possible phases
  7. The problem and an upper bound of $\frac{1}{2}2^k$ for adaptive algorithms
  8. An upper bound of $\frac{2}{3}2^k$ for non-adaptive algorithms
  9. A lower bound of $\frac{2}{3}2^k$ for non-adaptive algorithms
  10. A factor-2 advantage from larger phase-sets
  11. The general problem and an upper bound of $\frac{1}{2}2^k$ for adaptive algorithms
  12. An upper bound of $\frac{2^\ell}{2^\ell+1}2^k$ for non-adaptive algorithms
  13. A lower bound of $\frac{2^\ell}{2^\ell+1}2^k$ for non-adaptive algorithms
  14. Are bigger separations possible?
  15. Conclusion and discussion
  16. ...and 3 more sections

Figures (4)

  • Figure 1: The standard circuit for phase estimation
  • Figure 2: The function $f_1(x)$ (in blue) and the two functions that make its product form
  • Figure 3: $f_3(x)$ as defined in \ref{['eq:f3']}
  • Figure 4: The five functions in the product form for $f_3(x)$