Encoding of matrix product states into quantum circuits of one- and two-qubit gates

Abstract

Matrix product state (MPS) belongs to the most important mathematical models in, for example, condensed matter physics and quantum information sciences. However, to realize an N -qubit MPS with large N and large entanglement on a quantum platform is extremely challenging, since it requires high-level qudits or multi-body gates of two-level qubits to carry the entanglement. In this work, an efficient method that accurately encodes a given MPS into a quantum circuit with only one-and two-qubit gates is proposed. The idea is to construct the unitary matrix product operators that optimally disentangle the MPS to a product state. These matrix product operators form the quantum circuit that evolves a product state to the targeted MPS with a high fidelity. Our benchmark on the ground-state MPS's of the strongly-correlated spin models show that the constructed quantum circuits can encode the MPS's with much fewer qubits than the sizes of the MPS's themselves. This method paves a feasible and efficient path to realizing quantum many-body states and other MPS-based models as quantum circuits on the near-term quantum platforms.Matrix product state (MPS) is one of most successful mathematic tools in the contemporary physics. In condensed matter physics, MPS is the state ansatz behind the famous density matrix renormalization group (DMRG) algorithm [1, 2] and many of its variants [3][4][5][6][7]. MPS can efficiently describe the ground states and (purified) thermal states of one-dimensional (1D) gapped systems [8][9][10][11][12]. It has also been widely and successfully applied to other areas including statistic physics [13], non-equilibrium quantum physics [9, 14-17], field theories [18][19][20][21][22][23], machine learning [24][25][26][27][28], and so on.In particular, MPS is an important model in quantum information and computation (see, e.g., [29][30][31][32]). It can represent a large class of states, including GHZ [33] and AKLT states [34,35], which can implement non-trivial quantum computational tasks [36,37]. However, the realization of MPS on quantum hardwares is strictly limited. This is partially due to the fact that current techniques only permit short coherent time and small numbers of computing qubits. Solid progresses are reported in this direction recently, for instance, the realization of the GHZ state up to twenty qubits in a (relatively) long coherent time [38].However, MPS is hindered by another essential difficulty. There are two kinds of degrees of freedom in MPS, which are physical degrees of freedom corresponding to the Hilbert space in which the physical system is defined, and the virtual degrees that carry the entanglement of the MPS. In general, the dimension of the virtual degrees of freedom (denoted as χ) is much larger than the physical dimension (denoted as d). To realize an MPS in a quantum platform, one intuitively needs to realize χ-level qudits as the virtual degrees of freedom. This becomes almost impossible considering we usually take χ ∼ O(10 2 ) or even larger.Recently, an inspiring met...

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