[Paper Review] Reducing the qubit requirement of Jordan-Wigner encodings of $N$-mode, $K$-fermion systems from $N$ to $\lceil \log_2 {N \choose K} ceil$

Brent Harrison, Dylan Nelson+2 more|arXiv (Cornell University)|Nov 8, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a fermion-to-qubit encoding for N-mode, K-fermion systems that reduces the qubit requirement from N to ⌈log₂(N choose K)⌉, achieving the information-theoretic minimum. By constructing a permutation-based mapping from the Jordan-Wigner basis to a minimal qubit basis, the method leverages particle number conservation to eliminate redundant qubits, significantly improving resource efficiency for quantum simulations on near-term devices.

ABSTRACT

To simulate a fermionic system on a quantum computer, it is necessary to encode the state of the fermions onto qubits. Fermion-to-qubit mappings such as the Jordan-Wigner and Bravyi-Kitaev transformations do this using $N$ qubits to represent systems of $N$ fermionic modes. In this work, we demonstrate that for particle number conserving systems of $K$ fermions and $N$ modes, the qubit requirement can be reduced to the information theoretic minimum of $\lceil \log_2 {N \choose K} ceil$. This will improve the feasibility of simulation of molecules and many-body systems on near-term quantum computers with limited qubit number.

Motivation & Objective

• To reduce the qubit resource cost in quantum simulations of fermionic systems beyond existing mappings like Jordan-Wigner and Bravyi-Kitaev.
• To exploit particle number conservation in K-fermion, N-mode systems to minimize qubit requirements.
• To construct a fermion-to-qubit encoding that achieves the information-theoretic lower bound of ⌈log₂(N choose K)⌉ qubits.
• To demonstrate that such minimal encodings are feasible and can be implemented with low circuit depth using Clifford operations.

Proposed method

• Proposes a permutation-based mapping from the standard Jordan-Wigner basis to a new basis where qubit count is reduced to ⌈log₂(N choose K)⌉.
• Uses the fact that the number of distinct many-body Fock states in a K-fermion, N-mode system is (N choose K), requiring only log₂(N choose K) qubits to encode.
• Constructs a unitary transformation (a permutation of basis states) that maps the standard Fock basis to a new computational basis with fewer qubits.
• Ensures fermionic anticommutation relations are preserved through a modified Jordan-Wigner-like mapping in the new basis.
• Demonstrates that Clifford group operations (CNOTs and X gates) can implement the required permutations, enabling low-depth implementations.
• Shows that no Clifford permutation can reduce qubit count beyond one qubit less than the original N, limiting the maximum improvement to a single qubit in such cases.

Experimental results

Research questions

• RQ1 Can the qubit requirement for simulating K-fermion, N-mode systems be reduced below N qubits by exploiting particle number conservation?
• RQ2 Is it possible to construct a fermion-to-qubit encoding that achieves the information-theoretic minimum of ⌈log₂(N choose K)⌉ qubits?
• RQ3 What is the maximum qubit reduction achievable via Clifford group operations in such encodings?
• RQ4 How can the fermionic anticommutation relations be preserved in a minimal qubit basis?
• RQ5 What is the trade-off between qubit reduction and circuit depth in such encodings?

Key findings

• The qubit requirement for K-fermion, N-mode systems is reduced from N to ⌈log₂(N choose K)⌉, achieving the information-theoretic minimum.
• The proposed encoding is constructed via a permutation of the Fock basis that maps to a minimal qubit register, preserving fermionic statistics.
• The method achieves a qubit reduction of up to N − ⌈log₂(N choose K)⌉ qubits, which can be substantial for large N and small K.
• Clifford group operations (CNOTs and X gates) can implement the required permutations, enabling low-depth quantum circuits.
• No Clifford permutation can render more than one qubit redundant in the minimal basis, limiting the maximum qubit reduction via Clifford operations to one qubit.
• The construction is demonstrated explicitly for a 4-mode, 2-fermion system, confirming the feasibility and correctness of the approach.