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Superfast encodings for fermionic quantum simulation
Kanav Setia, Sergey Bravyi, Antonio Mezzacapo, James D. Whitfield
TL;DR
Fermionic quantum simulation requires encoding fermionic degrees of freedom into qubits, with the Superfast Encoding using O(md) qubits and producing Pauli operators of weight O(d). The paper proposes Generalized Superfast Encodings that retain the original encoding’s qubit count while changing error-correction properties or reducing simulator Pauli weight. One GSE corrects single-qubit errors under graph assumptions, whereas the original Superfast Encoding fails to do so for d ≤ 6; another GSE reduces Pauli weight from O(d) to O(log d).
Problem
Fermionic quantum simulation requires encoding fermionic degrees of freedom into qubits, with the Superfast Encoding using O(md) qubits and producing Pauli operators of weight O(d).
Source: p. 1Method
The paper proposes Generalized Superfast Encodings that retain the original encoding’s qubit count while changing error-correction properties or reducing simulator Pauli weight.
Source: p. 1Results
One GSE corrects single-qubit errors under graph assumptions, whereas the original Superfast Encoding fails to do so for d ≤ 6; another GSE reduces Pauli weight from O(d) to O(log d).
Source: pp. 1, 4Takeaways & Limitations
The GSE variants offer error correction without extra qubits under stated graph conditions and a lower-weight simulator Hamiltonian for the logarithmic-weight construction.
Source: p. 1The theorem guarantees correction for 3-connected graphs with even vertex degrees ≥6 and at most two edges per pair; any degree below 6 prevents correction, and dummy edges can meet conditions at slight qubit cost.
Source: p. 4
Abstract
from arXiv · showhide
Simulation of fermionic many-body systems on a quantum computer requires a suitable encoding of fermionic degrees of freedom into qubits. Here we revisit the Superfast Encoding introduced by Kitaev and one of the authors. This encoding maps a target fermionic Hamiltonian with two-body interactions on a graph of degree $d$ to a qubit simulator Hamiltonian composed of Pauli operators of weight $O(d)$. A system of $m$ fermi modes gets mapped to $n=O(md)$ qubits. We propose Generalized Superfast Encodings (GSE) which require the same number of qubits as the original one but have more favorable properties. First, we describe a GSE such that the corresponding quantum code corrects any single-qubit error provided that the interaction graph has degree $d\ge 6$. In contrast, we prove that the original Superfast Encoding lacks the error correction property for $d\le 6$. Secondly, we describe a GSE that reduces the Pauli weight of the simulator Hamiltonian from $O(d)$ to $O(\log{d})$. The robustness against errors and a simplified structure of the simulator Hamiltonian offered by GSEs can make simulation of fermionic systems within the reach of near-term quantum devices. As an example, we apply the new encoding to the fermionic Hubbard model on a 2D lattice.
Appendix A: Properties of the loop operators
Appendix A establishes algebraic properties of path and loop operators, then uses a spanning tree to show that independent loop operators generate the stabilizer group. The resulting code has m − 1 logical qubits.
Appendix A: Properties of the loop operators
- The appendix defines path operators as ordered products of edge operators along a path.A path may intersect itself, and a loop is a path whose endpoint equals its starting vertex.Source: Appendix A · p. 6
- Loop operators commute with all path operators and are Hermitian; reversing a path and composing paths obey the stated inverse and composition rules.The proof uses the even number of loop edges incident to any vertex subset to establish commutation.Source: Appendix A · p. 6
- A spanning tree reduces the loop-operator generators to those associated with edges outside the tree.Tree-edge loop operators are identity, so every loop operator is generated by the non-tree-edge loops.Source: Appendix A · p. 6
- The non-tree-edge loop generators are independent, yielding a stabilizer code with m − 1 logical qubits.The generators are independent because each has a distinct anticommuting operator; their number is |E| − |V| + 1.Source: Appendix A · p. 6
Appendix B: Local Majorana modes for vertex degree d ≥6
Appendix B constructs local Majorana modes that satisfy the error-correction condition for every even vertex degree d(i) ≥ 6. It gives examples and cyclic-shift constructions for higher degrees.
Appendix B: Local Majorana modes for vertex degree d ≥6
- For every even vertex degree d(i) ≥ 6, local Majorana modes can be chosen to satisfy the error-correction condition.The construction covers both cases where d(i)/2 is odd and where it is even.Source: Appendix B · p. 7
- For degree 8, the appendix gives an explicit assignment of local Majorana modes using four qubits.The listed modes are Pauli strings.Source: Appendix B · p. 7
- For degree 10, the appendix gives an explicit assignment of local Majorana modes using five qubits.The listed modes are Pauli strings.Source: Appendix B · p. 7
- For higher degrees, the construction obtains remaining modes through cyclic shifts of selected initial modes.The number of shifts depends on whether d(i)/2 is odd or even.Source: Appendix B · p. 7
Appendix C: Lack of error correction in the Superfast Encoding
Appendix C proves that the original Superfast Encoding fails to correct all single-qubit errors on regular graphs of degree d ≤ 6. Its argument derives necessary restrictions on edge labels, then counts singular edges to reach a contradiction.
Appendix C: Lack of error correction in the Superfast Encoding
- If the encoding corrected all single-qubit errors, edges with specified endpoint labels would have to be absent.The necessary condition is E1,p = Ed,p = ∅ for p ∈ {1, 2} and p ∈ {d − 1, d}.Source: Appendix C · p. 7
- For regular graphs of degree d ≤ 6, the Superfast Encoding does not correct all single-qubit errors.The proof establishes the cases d ≤ 5 and d = 6 separately.Source: Appendix C · p. 7
- Under that necessary condition, each vertex has at least two incident singular edges, and for degree d ≤ 5 the resulting count leads to a contradiction.An edge is singular if it belongs to E1,p or Ed,p for some p.Source: Appendix C · p. 7
- For degree 6, the assumed correction property forces exactly 2|V| singular edges, with four specified incident edges at every vertex.The proof then considers pairs among those edges and obtains a logical operator acting only on two qubits.Source: Appendix C · p. 7
Appendix D: 2D Hubbard Model
Appendix D describes the Superfast Encoding of a 2D Hubbard model and adds auxiliary modes to provide enough stabilizers for single-qubit error correction. With the chosen edge ordering, single-qubit errors in a unit cell have distinct syndromes.
Appendix D: 2D Hubbard Model
- The Hubbard model is represented using two square lattices for opposite spins, connected by vertical edges; lattice edges represent hopping operators and qubits occupy edges.The elementary loops correspond to stabilizers.Source: Appendix D · p. 8
- The original Superfast Encoding's stabilizers are insufficient to correct all single-qubit errors in the Hubbard model without auxiliary modes.The appendix notes that the original-encoding construction uses graph distance d = 8.Source: Appendix D · p. 8
- Each original-lattice plaquette receives a central auxiliary mode connected to its four vertices, creating four smaller independent stabilizer loops.The auxiliary modes add graph edges but have no fermionic terms in the target Hamiltonian; their vertex operators can be stabilizers.Source: Appendix D · p. 8
- With the specified edge ordering, the single-qubit errors in unit cell 6-7-11-10 have distinct syndromes.The encoding requires 12s qubits, where s is the number of sites in the original lattice.Source: Appendix D · p. 8
Appendix E: Fermionic operators in terms of edge operators
Appendix E lists edge-operator representations for several even fermionic operator types. It also includes a superconductivity interaction representation.
Appendix E: Fermionic operators in terms of edge operators
- The appendix identifies the number, excitation, number-excitation, Coulomb, and double-excitation operators as distinct fermionic operator types.Their expressions in terms of edge operators are given in Table I.Source: Appendix E · p. 9, Table I · p. 9
- The appendix also includes an edge-operator derivation for a superconductivity interaction.Source: Appendix E · p. 9
- Table I gives edge-operator representations for even fermionic operators.The listed types include number, excitation, number-excitation, Coulomb, and double-excitation operators.Source: Table I · p. 9, Appendix E · p. 9