Variational Quantum Computation of Excited States Oscar Higgott1,2, Daochen Wang1,3, and Stephen Brierley1 1 Riverlane, 3 Charles Babbage Road, Cambridge CB3 0GT 2 Department of Physics and Astronomy, University College London, London, WC1E 6BT 3Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742 July 1, 2019 The calculation of excited state energies of materials, or to understand some chemical reactions, electronic structure Hamiltonians has many im- such as those that involve photodissociation. However, portant applications, such as the calculation of classical methods such as density functional theory are optical spectra and reaction rates. While low- often unable to determine excited states, even for ma- depth quantum algorithms, such as the varia- terials where ground state energy calculations are pos- tional quantum eigenvalue solver (VQE), have sible. been used to determine ground state ener- Quantum computers have the potential to solve these gies, methods for calculating excited states cur- and other problems significantly faster than any known rently involve the implementation of high-depth methods using classical computers [1, 11, 19, 36]. How- controlled-unitaries or a large number of addi- ever many quantum algorithms will require quantum tional samples. Here we show how overlap esti- error correction, limiting their usefulness in the near mation can be used to deflate eigenstates once future [31]. Here we study hybrid quantum-classical al- they are found, enabling the calculation of ex- gorithms, which dramatically reduce the required gate cited state energies and their degeneracies. We depth to run and somewhat mitigate errors, by closely propose an implementation that requires the integrating classical and quantum subroutines [2,9, 14, same number of qubits as VQE and at most twice 15, 21, 23, 26, 40]. the circuit depth. Our method is robust to con- The variational quantum eigensolver (VQE), intro- trol errors, is compatible with error-mitigation duced in Ref. [30], is the first algorithm designed to strategies and can be implemented on near-term find the lowest eigenvalue of a Hamiltonian on a near- quantum computers. term, non-fault-tolerant quantum computer. VQE is based on the variational principle and utilises the fact 1 Introduction that quantum computers can store quantum states us- ing exponentially fewer resources than required classi- Eigenvalue problems are ubiquitous in almost all fields cally. VQE uses parameterised quantum circuits to pre- of science and engineering. Google's PageRank al- pare trial wavefunctions and compute their energy, and gorithm alone has had a significant impact on mod- a classical computer to find the parameters minimising ern society, and at its core solves an eigenvalue prob- this energy. The low circuit depth of VQE has led to the lem associated with a stochastic matrix describing the hope that it may enable near-term quantum-enhanced World Wide Web [28]. Another important example is computation. Principal Component Analysis (PCA) [13, 29], which Since its introduction, modifications have been sug- arXiv:1805.08138v5 [quant-ph] 28 Jun 2019 has widespread applications in bioinformatics, neuro- gested to enable VQE to find excited state ener- science, image processing, and quantitative finance. gies: e.g. a folded spectrum method [30] which re- The time-independent Schr¨odinger equation provides quires finding the expectation of the squared Hamil- yet another example of a fundamental eigenvalue prob- tonian with quadratically more terms, or symmetry- lem. Its numerical solution enables properties of atoms, based methods which are non-systematic [23]. Such molecules and materials to be predicted, with far- suggestions have been more recently superseded by reaching applications in materials design, drug discov- two proposals: a method that minimises the von Neu- ery and fundamental science [38]. Characterisation of mann entropy [35] and the quantum subspace expansion excited state energies of molecules is required to predict method [5, 24]. However, the von Neumann entropy charge and energy transfer processes in photovoltaic method (\WAVES") requires a large number of high- depth controlled-unitaries, and the quantum subspace Oscar Higgott: [email protected] expansion method requires a large number of additional Accepted in Quantum 2019-06-14, click title to verify 1 VARIATIONAL QUANTUM DEFLATION ALGORITHM (VQD) samples compared to VQE and introduces a new ap- QUANTUM CIRCUITS CLASSICAL CIRCUITS proximation. Expectation estimation i ) Our algorithm extends VQE to systematically find k of (λk) P1 (λk) λ h | | i ( excited states at almost no extra cost. We achieve this | i H 2 0 | | | Expectation estimation ) i ) k k of (λk) P2 (λk) λ by adding \overlap" terms onto the optimisation func- ( h | | i λ ( | ) ... i tion in order to exploit the fact that Hermitian ma- ⌘h λ ) to minimise ( k k Expectation estimation λ h λ ( | Classical adder calculates trices admit a complete set of orthogonal eigenvectors. i of (λk) Pn (λk) E h | | i β 1 from fiducial state − =0 k i Exploiting further the fact that VQE retains the clas- i ) P k λ ( Overlap estimation sical parameters of ansatz states that enable their re- 2 | | 2 )+ i of (λ ) (λ ) k 0 k ) λ |h | i| k ( λ ( preparation, low-depth quantum circuits can then be E | Overlap estimation ) i 2 = λ ) prepares readily used to calculate these overlap terms. of (λ1) (λk) ( k |h | i| λ h obj ( | Classical optimiser updates F i R β ... 1 − =0 k Overlap estimation i 2 P of (λk 1) (λk) Classical adder calculates 2 Variational quantum deflation algo- |h − | i| rithm <latexit sha1_base64="yR/JbkSxD+62BkpgO63Wfjq2Z7A=">AAAT4XiczVjdb9s2EFe7re2yr6Z53Au7uoCD2p6kNGuLQUWztF0DpF9pkhawU4OWaJuzRKok1SR1/Afsbdjr/rBtb/sn9jhgR0m2JX9EadAMM2CAPt7H7+54d6RboU+lMs0/z53/6ONPLly89OnCZ59/8eVXlxev7EoeCZfsuNzn4lULS+JTRnYUVT55FQqCg5ZPXrZ663r/5VsiJOVsWx2GZC/AHUbb1MUKSM3FxbChaO+dVIc+6UuFhZKKhwPkoLogrsKs45MKEjxiHvGQywUDZRUUUEaDKED71FNdZ8UNxqQuoZ2uciygKXKgkEuYIoJ4FeQJvO+0fOz2KqhNfd8B6tUVc28hA4Hy2LYSOCTvaAR6R0vkk7ZCMSLnlpndENpiumNZ5knhoSJ8LT8ikwBDwV0i5WSEPpBFLkDflE2PuFRnMDbqURxw5n0wkx1BCPs/eJnatM/I6HF+rpyRzVkH6L9PJhaC7ydl1aXAWr1buetIRbCvukNGovouZwpDGxHf1mI5p6/VQ/GPw+Jh2dV2KQM2JEno2CQYDBYaLdKhrK81hdRVkSCDOuMeQR50MMxc4thuAKZiWhkaVLiM6sN86/aisCLOnanKtcyaFkT90la54UNX83Czt1xCWgMWRKJSowfIG6Gkmf1BCbUFDyAiXuRS7COp1ae85qA0+H6IhByE1hhJBeEWf0vSduIg3o4NpVFPEN3UWTiUXdpWTtWureqfB8lPK0b64CDU8dLdFRGpaJAsQVep0RJ4CuuzpjXbhxxKO4uyRXy+DxodDX8uOlODOxUg+wSA2DxA9iSgU0FgBRCwQuVSEpirtdWrCaISYBodJTDb8D2u5Bi23M9meyWPm83A/RQmp4/DKczxIDrS0KdQmstHk6gbyXk6em2Xcljs2Vg0ysKkngKZdXJkvbnIZmX3FFj6vao1OBmeJNNxuOJMx/CKMg35xJ5HRL7LxMpHZZNpOlmXaiuZkq5atVurUxUPBtd9LCVconwU20GwciMf9EFLepBxqUHeRPQtmnnKHxWVGbh6jBfJOTkzLxoyCpp96piD13G2wAeicJOiNK9oZmLpcUlFGd+i0AM77zMFcg7eqd3OOgQOaof0WRmmvgZXZNVFR9W4hye/lrMu8xCOK5XgdoJF+zxEXUKKJyCAAZUeNvu89dPAyWb2xhkEKI6PHrqoHs/svWRWjl1J5lW1mk6uYm57xG2fgJuNuFkRd9JLNXe8Kua2R9yFSJIOlHL3prm17zWCpRrBTTLOuBiHKWGYJWsXydrzZVmRLJsjq6OUkR2W9lh2zDBL1i6SnYdZR7BItjff30wpJQqSWsm5nOWZZb9QxwQP6HA5Fx5l+tZWbmOhC2o5nQTxUYlFl2+UofBvVqrf1VZhIkyLxXU1lEuNJiZulPOCecw51mp1trAJkrA3MpOsY6SxyDTOVCK3FVvXHbE+un3rm7xy0ktyUueA6U2EmYoC6GC5yZiS01juE53neuYWW8nfwk14JHtIT8znO2tPtnceo/WNrfWdje0Xg3k4xqN02LMBjTvsopN4Rhvvj2h9c+3Fi431tc1iTGlshjNkfEQyWR9FTB9ChV05CTUmvj/M3bWtjbXtjadPAOgwiPcfPNyMaWht88enWxvbjx6j8u7z+8vaIGFe7mHUvHzNrJnxB00vrHRx7d71v//4yzCMZ83Fc68aHnejAJ59cXzrlhmqPXjWKer6BF5fEbzE4A2IO6QOS4YDIvf68X9DA3QdKB5qcwFfBm9HTc1K9HEg5WHQAk64vHXl5J4mztqrR6p9e69PWRgpwtzEUDvy9eTUfzRB7PTD0T+EBXYFBazI7WKBXXi75jXhlRCHROz1O4QHRInDnEv9ZIzmaTqgCUWvfAo84rAvu6BH1lI11IUHnS5peKHom00FhVxSfT+lrFOBgzRYgFxYk5GfXuzaNbiCWM8hKT8YyeeS8bXxjVE2LOOWcc94ZDwzdgx38ffFf65cuHJxyV36eemXpV8T1vPnUpklI/dZ+u1fnhbRUA==</latexit>sha1_base64="YP815qoE3qu0PCW3l0Rz70PuABw=">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