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Training Variational Quantum Algorithms Is NP-Hard — Even for Logarithmically Many Qubits and Free Fermionic Systems

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Training Variational Quantum Algorithms Is NP-Hard — Even for Logarithmically Many Qubits and Free Fermionic Systems Training variational quantum algorithms is NP-hard — even for logarithmically many qubits and free fermionic systems Lennart Bittel∗ and Martin Klieschy Heinrich Heine University Düsseldorf, Germany Variational quantum algorithms (VQAs) are proposed to solve relevant computational problems on near term quantum devices. Popular versions are variational quantum eigensolvers (VQEs) and quantum approximate optimization algorithms (QAOAs) that solve ground state problems from quantum chemistry and binary optimization problems, respectively. They are based on the idea to use a classical computer to train a parameterized quantum circuit. We show that the corresponding classical optimization problems are NP-hard. Moreover, the hardness is robust in the sense that for every polynomial time algorithm, there exists instances for which the relative error resulting from the classical optimization problem can be arbitrarily large, assuming P 6= NP. Even for classically tractable systems, composed of only logarithmically many qubits or free fermions, we show that the optimization is NP-hard. This elucidates that the classical optimization is intrinsically hard and does not merely inherit the hardness from the ground state problem. Our analysis shows that the training landscape can have many far from optimal persistent local minima. This means gradient and higher order decent algorithms will generally converge to far from optimal solutions. I. INTRODUCTION 0 U (φ ) | i 1 1 The last years have seen enormous progress towards large-scale quantum computation. A central goal of this 0 U3(φ3) effort is the implementation of a quantum computation | i UM that solves computational problems of practical relevance 0 U (φ ) faster than any classical computer. However, the noisy | i 5 5 nature of quantum gates and the high overhead cost of U2(φ2) noise reduction and error correction limit near term de- 0 U4(φ4) vices to shallow circuits [1]. | i Variational quantum algorithms (VQAs) have been proposed to bring us a step closer towards this goal. Here, φ an optimization problem is captured by a loss function guess Classical O(φ) Post given by expectation values of observables w.r.t. states h i φ Optimization Processing generated from a parameterized quantum circuit. Then a res classical computer trains the quantum circuit by optimiz- ing the expectation value over the circuit’s parameters. Figure 1. Sketch of a VQA optimization routine. This work Figure1 illustrates a possible VQA routine. Popular can- addresses the complexity of the classical optimization part didates to be used on near term devices are quantum (red). approximate optimization algorithms (QAOAs) [2] and variational quantum eigensolvers (VQEs) [3], see [4] for a review. optimal solution of relevant optimization problems (i.e. VQEs are proposed, for instance, to solve electronic arXiv:2101.07267v1 [quant-ph] 18 Jan 2021 the model mismatch is small). Second, the classical op- structure problems, which are central in quantum chem- timization over the parameters of the quantum circuit istry and material science. Proposals of QAOAs include needs to be solved fast enough and with sufficient accu- improved algorithms for quadratic optimization problems racy. We will focus on this second challenge. over binary variables such as the problem of finding the For the classical optimization several heuristic ap- MaxCut maximum cut of a graph ( ). For hybrid clas- proaches are known, which are mostly based on gradient sical/quantum computation to be successful, two chal- descent ideas and higher order methods. This is conve- lenges need to be overcome. First, one needs to find nient, as with the parameter shift rule [5], the gradient parameterized quantum circuits that have the expressive can be calculated efficiently. Methods include standard power to yield a sufficiently good approximation to the BFGS optimization and extensions [6] and natural gra- dient descent [7], which has a favorable performance at least for certain easy instances [8]. Second order methods ∗ [email protected] require a significant overhead in the number of measure- y [email protected] ments but can yield a better accuracy [9]. Quantum an- 2 alytic descent [10] uses certain classical approximations B. Notation of the objective function in order to reduce the number of quantum circuit evaluations at the cost of a higher We use the notation [n] := 1; : : : ; n . The Pauli ma- f g classical computation effort. trices are denoted by σx, σy, and σz. An operator X However, it has also been shown recently that there are acting on subsystem j of a larger quantum system is de- (j) (1) certain obstacles that need to be overcome to render the noted by X , e.g., σx is the Pauli-x-matrix acting on classical optimization successful. The training landscape subsystem 1. X refers to the operator norm of operator k k can have so-called barren plateaus where the loss function X. is effectively constant and hence yields a vanishing gradi- The number of edges of the graph with the adjacency ent, which prevents efficient training. This phenomenon matrix A is denoted by E(A) . By MaxCut(A) we denote j j can be caused, for example by random initializations [11] the solution of MaxCut for an adjacency matrix A, see and non-locality of the observable defining the loss func- Problem1. tion [12]. Also sources of randomness given by noise in Throughout, we only consider adjacency matrices A the gate implementations can cause similar effects [13]. of undirected unweighted graphs with at least one edge, Moreover, the problem of barren plateaus cannot be fully i.e., A 0; 1 d×d is a non-zero symmetric binary matrix 2 f g resolved by higher order methods [14]. with vanishing diagonal. In this work, we show that the existence of persistent local minima can also render the training of variational quantum algorithms infeasible. For this, we encode the II. A CONTINUOUS MaxCut OPTIMIZATION NP-hard MaxCut problem into the corresponding classical optimization for several versions of VQAs, which have We will introduce a continuous, trigonometric problem many far from optimal local minima. which we show to be NP-hard to optimize and approxi- Specifically, we obtain hardness results concerning the mate. This is related to earlier work about optimization optimization in four different settings: (i) We use an or- of trigonometric functions [17] for which NP-hardness is acle description of a quantum computer and show that known. For the specific class of functions, we also show the classical optimization of VQA is an NP-hard prob- the existence of an approximation ratio explicitly. We use lem, even if it only needs to be solved within constant this problem later to obtain hardness results for various relative precision. Next, we remove the oracle from the VQA versions. problem formulation by focusing on classically tractable systems where the underlying ground state problem is Problem 1 (MaxCut). efficiently solvable. Here, we consider quantum systems Instance: The adjacency matrix A 0; 1 d×d of an un- 2 f g where (ii) the Hilbert space dimension scales polynomi- weighted undirected graph. P ally in the number of parameters (i.e. logarithmically Task: Find S [n] that maximizes Ai;j. ⊂ i2S;j2[n]nS many qubits) or (iii) is composed of free fermions. (iv) If the setup is restricted to be of QAOA type, we show MaxCut is famously known to be NP-hard. Addition- that our hardness results also hold. ally MaxCut is APX-hard, meaning that every polyno- mial time approximation algorithm has an approxima- tion ratio Algorithm Solution α < 1 for at least some Optimal Solution ≤ max instances, assuming P = NP. It was shown that if 6 the unique games conjecture is true, then the best ap- A. Connection to complexity theory proximation ratio of a polynomial algorithm is αmax = θ/π min <θ<π 0:8786 [18], which is also what 0 (1−cos(θ)=2) ≈ The decision version of VQA optimization is in the the best known algorithms can guarantee [19]. Without 16 complexity class QCMA, problems that can be verified using this conjecture, it is proven that αmax ≤ 17 ≈ with a classical proof on a quantum computer. The 0:941 [20]. For our purpose we define a continuous, class QMA, which allows for the proof to be a quantum trigonometric version of MaxCut. Minima of real val- state, contains QCMA. Much about the relationship be- ued functions are given by real numbers that may not tween classical MA, QCMA and QMA is yet unknown. have an efficient numerical representation. However, it Prominently, finding the ground state energy of a local is common to say that a minimization problem is solved Hamiltonian is QMA-hard [15, 16]. This means that if if it is solved to exponential precision, which is the con- QCMA = QMA then VQA algorithms will not be able to vention we will also be using throughout this paper. The 6 solve the local Hamiltonian problem, but possibly other intuitive notion is that the hardness does not come from problems contained in QCMA. What our results also the difficulty of representing the minimum. show is that even if the relevant energy eigenstates are contained in the VQA ansatz, the classical optimization Problem 2 (Continuous MaxCut). may still be at least as difficult as solving NP problems Instance: The adjacency matrix A 0; 1 d×d of an un- 2 f g (Section IIIA). weighted graph. 3 Task: Find φ [0; 2π)d that minimizes The same minima (in the MaxCut formulation) are also 2 achieved by a greedy algorithm: start with a random bi- d 1 X partition of the vertex set, then repeatably change a sin- µ(φ) := Ai;j(cos(φi) cos(φj) 1): (1) 4 − gle vertex assignment if it increases the cut until the cut i;j=1 cannot be increased any further by this update rule.
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