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Physical Review A

Reachability deficit of variational Grover search

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

Published Jan 9, 2024• 8 citations
Publisher

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.

DOI
10.1103/physreva.109.012414

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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(n2)O({n}^{2})O(n2) 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.
The quantum approximate optimization algorithm~(QAOA) first proposed by Farhi et al. promises near-term applications based on its simplicity, universality, and provable optimality. A depth-p QAOA consists of p interleaved unitary transformations induced by two mutually non-commuting Hamiltonians. A long-standing question concerning the performance of QAOA is the dependence of its success probability as a function of circuit depth p. We make initial progress by analyzing the success probability of QAOA for realizing state transfer in a one-dimensional qubit chain using two-qubit XY Hamiltonians and single-qubit Hamiltonians. We provide analytic state transfer success probability dependencies on p in both low and large p limits by leveraging the unique spectral property of the XY Hamiltonian. We support our proof under a given QAOA ansatz with numerical optimizations of QAOA for up to NNN=20 qubits. We show that the optimized QAOA can achieve the well-known quadratic speedup, Grover speedup, over the classical alternatives. Treating QAOA optimization as a quantum control problem, we also provide numerical evidence of how the circuit depth determines the controllability of the QAOA ansatz.
The quantum approximate optimization algorithm (QAOA) is a quantum-classical hybrid algorithm intending to find the ground state of a target Hamiltonian. Theoretically, QAOA can obtain the approximate solution if the quantum circuit is deep enough. Actually, the performance of QAOA decreases practically if the quantum circuit is deep since near-term devices are not noise free, and the errors caused by noise accumulate as the quantum circuit increases. In order to reduce the depth of quantum circuits, we propose an ansatz dubbed as ``Shortcuts to QAOA'' (S-QAOA), S-QAOA provides shortcuts to the ground state of the target Hamiltonian by including more two-body interactions and releasing the parameter freedoms. To be specific, besides the existing ZZZZZZ interaction in the QAOA ansatz, other two-body interactions are introduced in the S-QAOA ansatz such that the approximate solutions could be obtained with smaller circuit depth. Considering the Max-Cut problem and the Sherrington-Kirkpatrick model, numerically computation shows the YYYYYY interaction has the best performance. The reason for this might arise from the counterdiabatic effect generated by YYYYYY interaction. On top of this, we release the freedom of parameters of two-body interactions, which a priori do not necessarily have to be fully identical, and numerical results show that it is worth paying the extra cost of having more parameter freedom since one has a greater improvement on success rate.
We study the Quantum Approximate Optimization Algorithm with a Grover mixer and independently sampled cost and mixing angles. Under a lattice condition on the cost values carried by the initial state, we prove a depth-independent lower bound for the variance of the loss at every depth, together with the same bound for the derivative with respect to the final mixing angle. The estimate is instance-dependent and, for fixed locality, is inverse polynomial in the number of qubits for integer-valued local objective functions with uniformly bounded local terms, in particular for MaxCut. We also establish Grover-type reachability bounds showing that the depth required to approximate a prescribed carried eigenspace is bounded below by a constant multiple of the inverse square root of its initial probability.
Variational quantum algorithms have become the de facto model for current quantum computations. A prominent example of such algorithms---the quantum approximate optimization algorithm (QAOA)---was originally designed for combinatorial optimization tasks, but has been shown to be successful for a variety of other problems. However, for most of these problems the optimal circuit depth remains unknown. One such problem is unstructured search, which consists of finding a particular bit string or, equivalently, preparing a state of high overlap with a target state. To bound the optimal QAOA depth for such a problem we build on its known solution in a continuous time quantum walk (CTQW). We Trotterize a CTQW to recover a QAOA sequence, and employ recent advances on the theory of Trotter formulas to bound the query complexity (circuit depth) needed to prepare a state that approaches perfect overlap with the target state. The obtained complexity exceeds Grover's algorithm complexity O(N12)O({N}^{\frac{1}{2}})O(N21​), but remains smaller than O(N12+c)O({N}^{\frac{1}{2}+c})O(N21​+c) for any c>0c>0c>0, which shows quantum advantage of QAOA over classical solutions. We verify our analytical predictions by numerical simulations of up to 68 qubits.
Variational quantum algorithms are the centerpiece of modern quantum programming. These algorithms involve training parametrized quantum circuits using a classical coprocessor, an approach adapted partly from classical machine learning. An important subclass of these algorithms, designed for combinatorial optimization on current quantum hardware, is the quantum approximate optimization algorithm (QAOA). Despite efforts to realize deeper circuits, experimental state-of-the-art implementations are limited to a fixed depth. However, it is known that problem density---a problem constraint to a variable ratio---induces underparametrization in fixed depth QAOA. Density-dependent performance has been reported in the literature, yet the circuit depth required to achieve fixed performance (henceforth called critical depth) remained unknown. Here, we propose a predictive model, based on a logistic saturation conjecture for critical depth scaling with respect to density. Focusing on random instances of MAX-2-SAT, we test our predictive model against simulated data with up to 15 qubits. We report the average critical depth, required to attain a success probability of 0.7, saturates at a value of 10 for densities beyond 4. We observe the predictive model to describe the simulated data within a $3\ensuremath{\sigma}$ confidence interval. Furthermore, based on the model, a linear trend for the critical depth with respect to problem size is recovered for the range of 5--15 qubits.
The quantum approximate optimization algorithm (QAOA) has become a cornerstone of contemporary quantum applications development. Here we show that the density of problem constraints versus problem variables acts as a performance indicator. Density is found to correlate strongly with approximation inefficiency for fixed depth QAOA applied to random graph minimization problem instances. Further, the required depth for accurate QAOA solution to graph problem instances scales critically with density. Motivated by Google's recent experimental realization of QAOA, we preform a reanalysis of the reported data reproduced in an ideal noiseless setting. We found that the reported capabilities of instances addressed experimentally by Google, approach a rapid fall-off region in approximation quality experienced beyond intermediate-density. Our findings offer new insight into performance analysis of contemporary quantum optimization algorithms and contradict recent speculation regarding low-depth QAOA performance benefits.