Tapering off qubits to simulate fermionic Hamiltonians

Motivation

• Fermionic simulation targets strongly interacting systems whose ground-state preparation and energy estimation are important but challenging for conventional and near-term quantum methods.Universal approaches require circuits beyond current demonstrations, motivating variational quantum algorithms.Source: §I · p. 1
• Particle-number and other symmetries allow simulation within restricted eigenspaces, potentially eliminating qubits without losing information.The paper seeks computationally efficient generalizations of reductions observed in molecular and Fermi-Hubbard examples.Source: §I · p. 1

Target system

• The target is an N-particle subspace of M fermionic modes, with Hamiltonians built from one- and two-body operators that conserve particle number.The energy objective is the minimum energy restricted to this fixed-particle subspace.Source: §I · p. 1

Encoding framework

• An encoding is an isometry from the target Hilbert space into Q-qubit states, whose image forms the simulator codespace.Encoded states are simulator states representing target states.Source: §I · p. 2

Simulator definition

• A valid simulator preserves the codespace, reproduces the target Hamiltonian there up to unitary equivalence, and contains a simulator ground state in that codespace.These conditions ensure equal ground energies and matching ground states modulo the encoding.Source: §I · p. 2

Sparsity and variational use

• The central design goal is Q < M qubits with sufficiently simple simulator Hamiltonians, especially r-sparse forms whose diagonal terms can be efficiently measured.This structure supports variational energy estimation through local basis changes and repeated measurements.Source: §I · p. 2

First-quantization encodings

• Q = N log2(M) qubits suffice in the low-particle regime N log2(M) < M, using a sparse first-quantization encoding.The construction also improves simulator sparsity with orthogonal arrays and introduces a sparse Hamiltonian enforcing antisymmetry.Source: Summary of results · p. 2

Second-quantization encodings

• Sparse second-quantization encodings can use LDPC codes with filling fraction ν, qubit-per-mode ratio η, relative distance 2ν, and encoding rate 1 − η.The required codes have column-sparse parity-check matrices and can approach the Gilbert–Varshamov bound.Source: Summary of results · p. 2
• Constant-fraction qubit elimination is possible for filling fraction ν < 1/4, while simulator sparsity scales with the number of nonzero target-Hamiltonian coefficients and ∥D_i∥ ≤ 1.The second-quantization construction therefore preserves sparsity up to proportionality in the target terms.Source: Summary of results · p. 2

Graph-based encodings

• Graph-based LDPC encodings give 2-sparse simulators for two-body operators, 32-sparse simulators for four-body operators, O(M^3) matrix-element computation, and eliminate approximately M/N qubits.Figure 1 gives numerically computed lower bounds on the number of modes M achievable for fixed N and Q.Source: Summary of results · pp. 2–3, Figure 1 · p. 3

Discrete symmetries

• For Z2 × Z2 parity symmetries, the construction uses Q = M − 2 qubits and more generally eliminates one qubit per detected Z2 symmetry.The method is illustrated on simple molecular-type Hamiltonians.Source: Summary of results · p. 3

Comparison

• The first-quantization encoding eliminates the most qubits but requires relatively small N, whereas second-quantization encodings apply more broadly and keep simulator and target term counts comparable up to constants.The preferred encoding depends on the target system; the first-quantization sparsity can nevertheless be as high as 9M^3.17.Source: Summary of results · p. 3

• The simulator Hilbert space uses Q qubits with computational basis states |s⟩, while Fock basis states are denoted |x⟩.Source: §II · p. 3
• The target observables considered are hermitian two-body or four-body fermionic operators.Source: §II · p. 3
• A qubit observable simulates a target observable when its codespace restriction is unitarily equivalent to the target action.Source: §II · p. 3
• An r-sparse simulator decomposes into r terms whose operators are diagonal in tensor-product bases, with the basis allowed to vary by term.Source: §II · p. 4

Encoding construction

• Q = mN qubits encode N occupied modes in N registers, with m = log2(M) when M is a power of two.Each register stores the binary representation of one mode index.Source: §III · p. 4
• The codespace is spanned by antisymmetric first-quantized states corresponding to N-particle Fock states.Source: §III · p. 4

Simulator Hamiltonian

• The simulator combines the first-quantized target Hamiltonian T + U with a penalty H⊥ for states outside the codespace.The penalty is constructed from register-exchange operators and has zero ground energy on the codespace.Source: §III · p. 4

Simulator validity

• Permutation symmetry ensures that the simulator preserves the codespace and is unitarily equivalent to the target Hamiltonian there.Source: §III · p. 4

Sparsity

• Orthogonal arrays group Pauli terms into common tensor-product bases, yielding sparsity r = 9^m = M^2 log2(3) ≈ M^3.17.Source: §III · p. 5

Ground-state protection

• The penalty coefficient can be chosen as g > 4N^-1∥U + T∥ = O(NM^4), ensuring the ground state lies in the codespace.The paper notes that this polynomial bound is conservative and may not be tight.Source: §III · p. 5

Penalty analysis

• Schur duality decomposes the simulator space into permutation and unitary-group sectors, enabling analysis of H⊥ eigenvalues.Source: §III · p. 5
• The smallest nonzero penalty eigenvalue is N/2 for a Young diagram with two columns of lengths N − 1 and 1.Source: §III · pp. 5–6

Encoding

• The encoding maps each occupation string x to a syndrome Ax, while simulator terms act on qubits according to the corresponding mode columns of A.The inverse set A^-1s contains weight-N strings consistent with syndrome s and is unique when A is N-injective.Source: §IV · p. 6
• N-injectivity makes the syndrome encoding isometric by uniquely identifying every weight-N occupation string.A parity-check matrix with minimum distance 2N+1 guarantees distinct syndromes for distinct weight-N strings.Source: §IV · p. 6

Sparsity

• Column sparsity of A yields sparse simulators, with matrix elements bounded by 1 and sparsity controlled exponentially by the maximum column weight.The construction gives sparsity 2^(2c(A)−1) and preserves bounded diagonal terms.Source: §IV · pp. 6–7

Observable simulation

• Fermionic observables are simulated by syndrome-conditioned bit flips and diagonal parity factors, reproducing the target observable exactly on the codespace.For the constructed simulator, OsimE=EOtgt, while its action outside the codespace is unrestricted.Source: §IV · p. 7

Hamiltonian construction

• The full Hamiltonian combines observable simulators with a codespace penalty, and choosing g∼M^4 ensures the ground state lies in the codespace.The sparsity is also bounded by a constant multiple of the number of nonzero target-Hamiltonian coefficients.Source: §IV · pp. 7–8

LDPC-code construction

• For fixed filling fraction ν=N/M, LDPC-based encodings use column-sparse parity checks with relative distance 2ν and encoding rate 1−η.The resulting qubit-per-mode ratio approaches η=h(2ν)+ϵ for suitable code families.Source: §IV · pp. 6, 8
• LDPC families approaching the Gilbert–Varshamov bound attain η=h(2ν)+ϵ, but their decoding constant grows as ϵ→0 and makes simulator sparsity impractical.The paper therefore turns to less optimal LDPC codes in later sections to address this trade-off.Source: §IV · p. 8

Decoding

• Efficient matrix-element evaluation reduces to minimum-weight decoding, which is generally NP-hard but is linear-time for special LDPC families and O(M^3) for the discussed high-girth constructions.A finite-size algorithm is also available for M≤50, although it is not asymptotically efficient.Source: §IV · p. 8

Bipartite structure

• A bipartite parity-check matrix partitions qubits into L and R so every column has odd overlap with both parts.This structure stabilizes the codespace through Z(L) and Z(R), whose eigenvalues are fixed by the particle number.Source: §V · p. 8
• The bipartite encoding has sparsity parameters determined by the reduced column-overlap structure of A.The paper states the resulting parameters after introducing the bipartite condition.Source: §V · p. 8

Sparsity reduction

• For two-body observables, the modified simulator retains exactly the same action on the codespace as the original simulator.Multiplying terms by stabilizer-dependent factors changes the orthogonal-complement action without changing the encoded observable.Source: §V · p. 8
• When the relevant columns overlap, stabilizer factors cancel selected Pauli-Z contributions and reduce the simulator sparsity to 2^k−3≤2^(2c(A)−3).The cancellation uses Z(L), Z(R), and their product according to the selected qubit pair.Source: §V · pp. 8–9

Graph-based encoding

• A bipartite graph with Q vertices and M edges yields an encoding E|x⟩=|Ax⟩ whose few-body simulators have r2=2 and r4=32.The incidence matrix A has one row per vertex and one column per edge.Source: §VI · p. 9

Injectivity and girth

• N-injectivity holds exactly when the bipartite graph has girth g≥2N+2.The equivalence follows because collisions between weight-N strings correspond to cycles of length at most 2N.Source: §VI · p. 9

Graph optimization

• Maximizing the number of encodable modes for fixed Q and N reduces to finding the largest bipartite graph with girth at least 2N+2.Greedy graph-construction algorithms provide lower bounds on the achievable number of modes.Source: §VI · p. 9

Explicit constructions

• A cycle-and-chord construction has girth 2N+2 and gives Q=M−M/(2+N) for N≤M/2−O(1).The construction uses an even cycle and chords whose paths contain N edges.Source: §VI · p. 9

Qubit reduction

• The graph-based family eliminates approximately M/N qubits; one example encodes M=25 modes with N=3 particles into Q=20 qubits.The example graph has girth g=8.Source: §VI · p. 9

Encoding locality

• Binary-tree encoding maps few-body fermionic observables to Pauli-like operators acting on O(log M) qubits, unlike Jordan-Wigner and parity mappings, which can act on all M qubits.This locality advantage remains relevant after symmetry-based qubit removal.Source: §VII · p. 10

Particle and spin conservation

• For molecular Hamiltonians conserving spin-up and spin-down particle numbers, the binary-tree simulator has two single-qubit symmetries that permit removing two qubits.The relevant rows encode the total and spin-resolved parities.Source: §VII · pp. 10–11

Qubit elimination

• Replacing the corresponding σz operators by their fixed eigenvalues removes qubits M and M/2 without changing the simulator’s eigenvalue problem.The reduced Hamiltonian acts on M−2 qubits.Source: §VII · p. 11

Parity encoding

• The same two-qubit removal argument applies to parity encoding because its relevant binary-matrix rows coincide with those of the binary-tree encoding.The removal preserves the stated locality properties for the binary-tree construction.Source: §VII · p. 11

Symmetry groups

• A symmetry group is an abelian subgroup of Pauli operators whose elements commute with every simulator term and exclude −I.Independent generators can be transformed by a Clifford unitary into single-qubit Z operators.Source: §VIII · p. 11

Trade-off

• Clifford conjugation preserves the number of Pauli terms but can increase their support, so reduced Hamiltonians may have higher-weight terms.This is the main practical trade-off of the general multiple-symmetry procedure.Source: §VIII · p. 11

Systematic search

• The symmetry search represents Pauli terms as binary vectors and computes the kernel of a parity-check matrix to obtain candidate commuting operators.A maximal abelian subgroup of the kernel-generated Pauli group supplies independent symmetry generators.Source: §VIII · p. 11

Qubit removal

• After a Clifford transformation, k independent symmetries become X-type operators on k designated qubits, whose eigenvalues can be substituted to remove those qubits.The construction uses single-qubit operators that anticommute with their matching symmetry and commute with the others.Source: §VIII · p. 12

Molecular examples

• The molecular study applies this symmetry search to H2 and small molecules using Jordan-Wigner, parity, and binary-tree mappings with STO-3G integrals.The resulting symmetry generators and matching single-qubit operators are reported in Table I.Source: §VIII · p. 12

Jordan-Wigner mapping

• Jordan-Wigner encoding stores each mode occupation in an ordered qubit and uses preceding-mode parity checks, producing Pauli operators supported on O(M) qubits.Its action flips one bit while adding a sign determined by preceding occupied modes.Source: Appendix A · p. 12

Parity mapping

• Parity mapping stores cumulative occupation parity and, like Jordan-Wigner, can produce Pauli strings with support O(M).The mapping is described as dual to Jordan-Wigner.Source: Appendix A · p. 12

Binary-tree mapping

• Binary-tree mapping improves locality to O(log(M))-local terms.Its update, parity, flip, and remainder sets can be generated recursively from the binary-string partial order.Source: Appendix A · pp. 12–13

Operator construction

• The update, parity, and flip sets determine the binary-tree operator construction and its support structure.The remainder set is obtained as R(j)=P(j)\F(j).Source: Appendix A · p. 12

Appendix B: The hydrogen molecule

• The simulator Hamiltonian initially contains M = 4 qubits and r = 14 Pauli terms.Source: Appendix B · p. 13
• The Pauli terms are encoded as columns of an 8 × 14 binary matrix, from which the parity-check matrix is constructed by Gaussian elimination.Source: Appendix B · p. 13
• Three linearly independent kernel vectors generate the symmetries used for tapering.The selected vectors are (0000|1100), (0000|1010), and (0000|1001).Source: Appendix B · p. 13
• A Clifford transformation conjugates the Pauli operators and produces a transformed simulator Hamiltonian whose relevant operators act on qubits 2, 3, and 4 only through σx or I.Source: Appendix B · p. 14
• Three qubits are removed from the four-qubit H2 Hamiltonian, reducing it to a trivial single-qubit problem.The operators acting on qubits 2, 3, and 4 can be replaced by eigenvalues ±1.Source: Appendix B · pp. 13–14

Appendix C: Decoding algorithm

• A is N-injective exactly when ker(A) contains no vector in W(M, 2K) for any 1 ≤ K ≤ N.Source: Appendix C · p. 14
• W(M, N) denotes the set of all M-bit strings with weight N.Source: Appendix C · p. 14
• The criterion applies to a fixed N-injective matrix A of size Q × M.Source: Appendix C · p. 14