Reachability deficit of variational Grover search

Xiao-Wei Li, Xiao-Ming Zhang, Bin Cheng, and Man-Hong Yung

Phys. Rev. A 109, 012414 – Published 9 January 2024

Abstract

The quantum approximate optimization algorithm (QAOA) is promising for achieving quantum computational advantage with near-term quantum devices. It was numerically shown that the QAOA cost functions exhibit a phenomenon called reachability deficit (RD), where the success probability cannot reach unity until the circuit depth exceeds a certain critical value. However, an in-depth theoretical understanding of RD remains lacking. Here we focus on a variational variant of Grover search on multiple marked solutions as a prototype for analyzing the RD problem, where we further relax the criterion of reachability by tolerating a certain probability of failure. Specifically, we obtain a general analytical expression relating the critical depth of the quantum circuit to the solution density. In the dilute limit, the critical depth is consistent with the Grover bound, exhibiting a robust quadratic scaling that is insensitive to the failure probability. Moreover, we also find that the projective mixing Hamiltonian performs significantly better than the traditional mixing Hamiltonian in the QAOA, although it is less favorable in terms of physical implementation. However, by taking into account two-body interactions in the mixing Hamiltonian, the performance becomes on par with the projective mixing Hamiltonian at the cost of $O (n^{2})$ additional terms. These results represent a simplified but insightful model of the QAOA, fully addressing the dependence of required circuit depth over the ground-state degeneracy.

Received 25 September 2023
Accepted 21 December 2023

DOI:https://doi.org/10.1103/PhysRevA.109.012414

Physics Subject Headings (PhySH)

Quantum algorithms Search strategy

Quantum Information, Science & Technology

Authors & Affiliations

Xiao-Wei Li ¹, Xiao-Ming Zhang^2,3, Bin Cheng^2,4, and Man-Hong Yung^1,5,6,*

¹Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
²Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
³Center on Frontiers of Computing Studies, Peking University, Beijing 100871, China
⁴Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, New South Wales 2007, Australia
⁵Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
⁶Shenzhen Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China

^*yung@sustech.edu.cn

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Vol. 109, Iss. 1 — January 2024

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Figure 1
Critical depth $p_{ɛ}^{*} (α)$ versus the problem hardness $α$ . Here $ɛ$ is the required failure probability. The dash-dotted line, dotted line, and solid line represent the analytical results for $ɛ$ values of $10^{- 1}$ , $10^{- 2}$ , and $10^{- 3}$ respectively. The pentagons, diamonds, and circles correspond to the numerical results for the three required failure probabilities. The horizontal axis indicated by the arrow is the critical density $α_{c}$ . The inset is a close-up of the main panel.
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Figure 2
Reachability deficit of variational Grover search. (a) Numerical simulation of the minimum cost function $C_{p}^{*}$ versus problem hardness $α$ . As the problem hardness increases, the minimum cost function monotonically increases. (b) Numerical simulation results of the minimum cost function $C_{p}^{*}$ as a function of circuit depth $p$ . The results show that as the depth increases, the minimum cost function monotonically decreases. In addition, for a given problem hardness, there exists a critical depth such that when the circuit depth exceeds the critical depth, the minimum cost function $C_{p}^{*} = 0$ can be achieved.
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Figure 3
Numerical simulation of the reachability deficit for the first-order and second-order approximate mixing Hamiltonian. (a) Minimum cost function $C_{p}^{*}$ versus circuit depth $p$ . (b) Critical depth $p^{*} (α)$ vs problem hardness $α$ . The blue solid line represents the case where the mixing Hamiltonian is $H_{B} = {(| + 〉 〈 + |)}^{\otimes n}$ and the red dash-dotted and green dotted lines represent the cases of first-order and second-order approximate mixing Hamiltonians, respectively. We simulate a six-qubit system with a problem hardness $α = 6$ in (a). The results show that a higher-order approximate mixing Hamiltonian may have better performance, with $H_{B}$ having the best performance. The upper and lower edges of the shaded area in (b) are given by expressions of $p_{u p}$ and $p_{l o w}$ in Eqs. (28) and (29), respectively. Note that the first-order approximation asymptotically conforms to the upper boundary numerically, while the second-order approximation approaches the behavior in the case of $H_{B}$ .
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Figure 4
Trajectory of the states in each depth for different optimal algorithms. (a) Phase-matching Grover search (Long's algorithm), In this scheme, the trajectory of the quantum state does not lie on the geodesic of the Bloch sphere, but the arc length for each step is uniform. (b) Brassard's algorithm, whose parameter setting is given in the main text. In this scheme, the trajectory lies on the geodesic of the Bloch sphere. For the first $p - 1$ steps, the arc length for each step is $2 β$ , while the arc length for the final step is determined by $π - (2 p + 1) β$ to ensure that the final trajectory lands on the south pole of the Bloch sphere. (c) Equal interval geodesic scheme. In this scheme, the trajectory lies on the geodesic of the Bloch sphere and the arc length for each step is $(π - β) / p$ .
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Figure 5
Construction of unitary operators (a) $V_{C} (ϕ)$ and (b) $V_{B} (θ)$ . (a) $V_{C} (ϕ) = (I_{2^{n} - 2^{m}}) \oplus (e^{i ϕ} I_{2^{m}})$ . The first $m$ qubits remain idle, while the subsequent $α$ qubits are subjected to a controlled phase shift gate. (b) $V_{B} (θ) = H^{\otimes n} (e^{i θ} \oplus I_{2^{n} - 1}) H^{\otimes n}$ . For details see Refs. [11, 25].
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