Variational Quantum State Diagonalization Ryan LaRose,1, 2 Arkin Tikku,1, 3 Étude O’Neel-Judy,1 Lukasz Cincio,1 and Patrick J. Coles1 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 2Department of Computational Mathematics, Science, and Engineering & Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48823, USA. 3Department of Physics, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom Variational hybrid quantum-classical algorithms are promising candidates for near-term imple- mentation on quantum computers. In these algorithms, a quantum computer evaluates the cost of a gate sequence (with speedup over classical cost evaluation), and a classical computer uses this information to adjust the parameters of the gate sequence. Here we present such an algorithm for quantum state diagonalization. State diagonalization has applications in condensed matter physics (e.g., entanglement spectroscopy) as well as in machine learning (e.g., principal component analy- sis). For a quantum state ρ and gate sequence U, our cost function quantifies how far UρU y is from being diagonal. We introduce novel short-depth quantum circuits to quantify our cost. Minimizing this cost returns a gate sequence that approximately diagonalizes ρ. One can then read out approx- imations of the largest eigenvalues, and the associated eigenvectors, of ρ. As a proof-of-principle, we implement our algorithm on Rigetti’s quantum computer to diagonalize one-qubit states and on a simulator to find the entanglement spectrum of the Heisenberg model ground state. I. INTRODUCTION the classical computer uses this cost information to adjust the parameters of the gate sequence. Variational hybrid The future applications of quantum computers, assum- algorithms have been proposed for Hamiltonian ground ing that large-scale, fault-tolerant versions will eventually state and excited state preparation [8, 14, 15], approx- be realized, are manifold. From a mathematical perspec- imate optimization [7], error correction [16], quantum tive, applications include number theory [1], linear alge- data compression [17, 18], quantum simulation [19, 20], bra [2–4], differential equations [5,6], and optimization and quantum compiling [21]. A key feature of such algo- [7]. From a physical perspective, applications include rithms is their near-term relevance, since only the sub- electronic structure determination [8,9] for molecules routine of cost evaluation occurs on the quantum com- and materials and real-time simulation of quantum dy- puter, while the optimization procedure is entirely clas- namical processes [10] such as protein folding and photo- sical, and hence standard classical optimization tools can excitation events. Naturally, some of these applications be employed. are more long-term than others. Factoring and solv- In this work, we consider the application of diagonaliz- ing linear systems of equations are typically viewed as ing quantum states. In condensed matter physics, diago- longer term applications due to their high resource re- nalizing states is useful for identifying properties of topo- quirements. On the other hand, approximate optimiza- logical quantum phases—a field known as entanglement tion and the determination of electronic structure may spectroscopy [22]. In data science and machine learning, be nearer term applications, and could even serve as diagonalizing the covariance matrix (which could be en- demonstrations of quantum supremacy in the near fu- coded in a quantum state [2, 23]) is frequently employed ture [11, 12]. for principal component analysis (PCA). PCA identifies A major aspect of quantum algorithms research is to features that capture the largest variance in one’s data make applications of interest more near term by reducing and hence allows for dimensionality reduction [24]. quantum resource requirements including qubit count, arXiv:1810.10506v2 [quant-ph] 26 Jun 2019 Classical methods for diagonalization typically scale circuit depth, numbers of gates, and numbers of mea- polynomially in the matrix dimension [25]. Similarly, surements. A powerful strategy for this purpose is al- the number of measurements required for quantum state gorithm hybridization, where a fully quantum algorithm tomography—a general method for fully characterizing is turned into a hybrid quantum-classical algorithm [13]. a quantum state—scales polynomially in the dimension. The benefit of hybridization is two-fold, both reducing Interestingly, Lloyd et al. proposed a quantum algo- the resources (hence allowing implementation on smaller rithm for diagonalizing quantum states that can poten- hardware) as well as increasing accuracy (by outsourcing tially perform exponentially faster than these methods calculations to “error-free” classical computers). [2]. Namely, their algorithm, called quantum princi- Variational hybrid algorithms are a class of quantum- pal component analysis (qPCA), gives an exponential classical algorithms that involve minimizing a cost func- speedup for low-rank matrices. qPCA employs quantum tion that depends on the parameters of a quantum gate phase estimation combined with density matrix exponen- sequence. Cost evaluation occurs on the quantum com- tiation. These subroutines require a significant number puter, with speedup over classical cost evaluation, and of qubits and gates, making qPCA difficult to implement 2 in the near term, despite its long-term promise. free to increase m with increased algorithmic complex- Here, we propose a variational hybrid algorithm for ity (discussed below). The outputted eigenvalues will be quantum state diagonalization. For a given state ρ, our in classical form, i.e., will be stored on a classical com- algorithm is composed of three steps: (i) Train the pa- puter. In contrast, the outputted eigenvectors will be in rameters α of a gate sequence Up(α) such that ρ~ = quantum form, i.e., will be prepared on a quantum com- y Up(αopt)ρUp(αopt) is approximately diagonal, where puter. This is necessary because the eigenvectors would n αopt is the optimal value of α obtained (ii) Read out have 2 entries if they were stored on a classical com- the largest eigenvalues of ρ by measuring in the eigen- puter, which is intractable for large n. Nevertheless, one basis (i.e., by measuring ρ~ in the standard basis), and can characterize important aspects of these eigenvectors (iii) Prepare the eigenvectors associated with the largest with a polynomial number of measurements on the quan- eigenvalues. We call this the variational quantum state tum computer. diagonalization (VQSD) algorithm. VQSD is a near-term Similar to classical eigensolvers, the VQSD algorithm algorithm with the same practical benefits as other vari- is an approximate or iterative diagonalization algorithm. ational hybrid algorithms. Employing a layered ansatz Classical eigenvalue algorithms are necessarily iterative, for Up(α) (where p is the number of layers) allows one not exact [26]. Iterative algorithms are useful in that to obtain a hierarchy of approximations for the eigeval- they allow for a trade-off between run-time and accuracy. ues and eigenvectors. We therefore think of VQSD as an Higher degrees of accuracy can be achieved at the cost of approximate diagonalization algorithm. more iterations (equivalently, longer run-time), or short We carefully choose our cost function C to have the run-time can be achieved at the cost of lower accuracy. following properties: (i) C is faithful (i.e, it vanishes if This flexibility is desirable in that it allows the user of the and only if ρ~ is diagonal), (ii) C is efficiently computable algorithm to dictate the quality of the solutions found. on a quantum computer, (iii) C has operational meanings The iterative feature of VQSD arises via a layered such that it upper bounds the eigenvalue and eigenvector ansatz for the diagonalizing unitary. This idea similarly error (see Sec. IIA), and (iv) C scales well for training appears in other variational hybrid algorithms, such as purposes in the sense that its gradient does not vanish the Quantum Approximate Optimization Algorithm [7]. exponentially in the number of qubits. The precise defi- Specifically, VQSD diagonalizes ρ by variationally updat- nition of C is given in Sec.IIA and involves a difference ing a parameterized unitary Up(α) such that of purities for different states. To compute C, we intro- y duce novel short-depth quantum circuits that likely have ρ~p(α) := Up(α)ρUp (α) (1) applications outside the context of VQSD. To illustrate our method, we implement VQSD on is (approximately) diagonal at the optimal value αopt. Rigetti’s 8-qubit quantum computer. We successfully (For brevity we often write ρ~ for ρ~p(α).) We assume a diagonalize one-qubit pure states using this quantum layered ansatz of the form computer. To highlight future applications (when larger quantum computers are made available), we implement Up(α) = L1(α1)L2(α2) ··· Lp(αp) : (2) VQSD on a simulator to perform entanglement spec- Here, p is a hyperparameter that sets the number of layers troscopy on the ground state of the one-dimensional (1D) L (α ), and each α is a set of optimization parameters Heisenberg model composed of 12 spins. i i i that corresponds to internal gate angles within the layer. Our paper is organized as follows. Section II outlines The parameter α in (1) refers to the collection of all α the VQSD algorithm and presents its implementation. i for i = 1; :::; p. Once the optimization procedure is fin- In Sec. III, we give a comparison to the qPCA algo- ished and returns the optimal parameters α , one can rithm, and we elaborate on future applications. Section opt then run a particular quantum circuit (shown in Fig.1(c) IV presents our methods for quantifying diagonalization and discussed below) N times to approximately de- and for optimizing our cost function. readout termine the eigenvalues of ρ. The precision (i.e, the num- ber of significant digits) of each eigenvalue increases with N and with the eigenvalue’s magnitude. Hence for II. RESULTS readout small Nreadout only the largest eigenvalues of ρ will be precisely characterized, so there is a connection between A.