Tapering off qubits to simulate fermionic Hamiltonians (1701.08213v1)

Abstract: 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.

• The paper introduces symmetry-based encoding methods that significantly reduce qubit requirements for simulating fermionic many-body systems.
• It demonstrates both first and second quantization approaches, leveraging U(1) symmetry and LDPC codes to enhance variational quantum algorithm performance.
• Results show improved Hamiltonian sparsity and qubit efficiency, enabling the simulation of larger molecular and material systems on current quantum devices.

Tapering off Qubits to Simulate Fermionic Hamiltonians

The paper "Tapering off qubits to simulate fermionic Hamiltonians" explores efficient methods of simulating fermionic many-body systems on quantum computers by reducing the qubit requirements necessary for such simulations. The researchers propose innovative encodings that leverage symmetries in fermionic Hamiltonians, presenting them in both theoretical and practical contexts.

The authors begin by examining the reduction of degrees of freedom inherent in fermionic systems due to symmetries. They focus on variational quantum algorithms and demonstrate the advantage of certain encodings associated with a $U(1)$ symmetry, which requires far fewer qubits than standard approaches. Specifically, they discuss how a system with $M$ modes and $N$ particles can be encoded using $Q=N\log_2(M)$ qubits based on a first quantization framework. They proceed to propose an enhanced version of this encoding to facilitate variational quantum algorithms, also showing improvements in Hamiltonian sparsity through the use of orthogonal arrays.

Looking at a second quantization approach, the paper suggests using classical Low-Density Parity-Check (LDPC) codes to create encodings with notable qubit efficiency. These encodings are characterized by a specific filling fraction $\nu=N/M$ and qubit-per-mode ratio $\eta=Q/M<1$. Derived from the properties of classical LDPC codes, the sparse encodings exploit graph structures, enabling the removal of a substantial fraction of qubits—up to $M/N$—from the quantum simulation without losing vital information.

The authors also delve into $Z_2$ and multiple $Z_2$ symmetry-based qubit elimination strategies. Here, the combination of symmetry groups with classical error-correcting code properties leads to efficient state encoding in reduced qubit spaces. The work showcases systematic methods for detecting these symmetries and explores their implications using practical examples like molecular Hamiltonians.

The implications of these findings are considerable for both practical quantum computer implementations and theoretical quantum studies. Reducing the number of required qubits expands the potential for simulating larger, more complex fermionic systems on existing quantum devices, offering a more efficient pathway to leveraging quantum computation in quantum chemistry and material science scenarios. Future research could further explore the encoding sparsity achievable under higher filling fractions and tailor these methods to other symmetry types in fermionic systems.