Tapering off qubits to simulate fermionic Hamiltonians

By Sergey Bravyi, Jay M. Gambetta, Antonio Mezzacapo, Kristan Temme. 2017

339 citations recorded by the source corpus.

We discuss encodings of fermionic many-body systems by qubits in the presence of symmetries. Such encodings eliminate redundant degrees of freedom in a way that preserves a simple structure of the system Hamiltonian enabling quantum simulations with fewer qubits. First we consider $U(1)$ symmetry describing the particle number conservation. Using a previously known encoding based on the first quantization method a system of $M$ fermi modes with $N$ particles can be simulated on a quantum computer with $Q=N\log{(M)}$ qubits. We propose a new version of this encoding tailored to variational quantum algorithms. Also we show how to improve sparsity of the simulator Hamiltonian using orthogonal arrays. Next we consider encodings based on the second quantization method. It is shown that encodings with a given filling fraction $\nu=N/M$ and a qubit-per-mode ratio $\eta=Q/M<1$ can be constructed from efficiently decodable classical LDPC codes with the relative distance $2\nu$ and the encoding rate $1-\eta$. A family of codes based on high-girth bipartite graphs is discussed. Graph-based encodings eliminate roughly $M/N$ qubits. Finally we consider discrete symmetries, and show how to eliminate qubits using previously known encodings, illustrating the technique for simple molecular-type Hamiltonians.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/85aa170f21b66e1cd35b9fc893d00a074d048c11

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Citations in the corpus

Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator (2018, methodology): “This reduction of the problem for the hydrogen molecule was first noted in [50], developed into a general method in [68] and used in superconducting implementations of several problems in [63].”
Error mitigation extends the computational reach of a noisy quantum processor (2018, methodology): “We map the interacting fermion problem for H2 and LiH on two and four qubits respectively, taking advantage of fermionic spin-parity symmetries [27] and the freezing of core-shell orbitals, as detailed in [9].”
Computation of Molecular Spectra on a Quantum Processor with an Error-Resilient Algorithm (2017, methodology): “[30] Bravyi S, Gambetta JM, Mezzacapo A, Temme K (2017) Tapering off qubits to simulate fermionic hamiltonians.”
Low-Depth Quantum Simulation of Materials (2017, background): “[27] Sergey Bravyi, Jay M Gambetta, Antonio Mezzacapo, and Kristan Temme, “Tapering off qubits to simulate fermionic Hamiltonians,” arXiv:1701.”
Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm (2019, background): “We note that there are many choices of qubit mappings and unitary transformations that can be used to reduce the complexity of molecular simulations.(37) However, we have formulated our ansätze to preserve symmetries in the natural basis where each qubit represents a physical spin orbital.”
Robust data encodings for quantum classifiers (2020, background): “, the reduction in the number of qubits in quantum chemistry applications due to qubit tapering [78].”
VQE method: a short survey and recent developments (2021, methodology): “To evaluate the energy on a quantum computer, the Hamiltonian for indistinguishable fermions has to bemapped onto the Hamiltonian of distinguishable qubits by using one of three common mappings: Jordan–Wigner (Jordan and Wigner 1928), parity (Bravyi et al. 2017), or Bravyi–Kitaev (Bravyi and Kitaev ”
Programmable quantum simulations of spin systems with trapped ions (2019, neutral): “First-quantization mapping can also been used (Babbush et al., 2017; Bravyi et al., 2017).”