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Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

Zhang Jiang, Amir Kalev, Wojciech Mruczkiewicz, Hartmut Neven

arXiv:1910.10746v2quant-ph

TL;DR

Learning reduced density matrices is useful in quantum simulation, but fermionic operators do not generally permit large groups of commuting measurements. The paper introduces a ternary-tree fermion-to-qubit mapping and a Bell-basis measurement method, then uses them to estimate fermionic RDM elements with one repeated circuit. The stated repetition bound depends on n and k, while the qubit RDM method's bound is independent of system size.

  • Problem

    Learning k-particle RDMs is useful for estimating energies and deriving system properties, while noncommuting fermionic operators complicate simultaneous measurement.

    Source: §1 · p. 1, §4 · p. 5

  • Method

    The paper combines a ternary-tree fermion-to-qubit mapping with Bell-basis measurements to estimate RDM elements using a single circuit.

    Source: §1 · pp. 1–2, §4 · p. 5

  • Results

    ≲(2n + 1)^k/ϵ^2 repetitions suffice for k-fermion RDM elements, versus ≲(n + 1)^(k log2 3)/ϵ^2 for the Bravyi-Kitaev transformation.

    Source: §4 · p. 5

  • Takeaways & Limitations

    The qubit RDM method is suited to quantum computers with slow circuit-updating rates and is informationally complete, allowing reconstruction of the entire quantum state.

    Source: §4 · p. 5

    With finite measurement rounds, RDMs involving fewer qubits are estimated more accurately than those involving more qubits.

    Source: §3.1 · p. 4

Abstract

from arXiv · show

We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an $n$-mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on $\lceil \log_3(2n+1)\rceil$ qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less than $\log_3(2n)$ qubits on average. We apply it to the problem of learning $k$-fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that using the ternary-tree mapping one can determine the elements of all $k$-fermion RDMs, to precision $\epsilon$, by repeating a single quantum circuit for $\lesssim (2n+1)^k \epsilon^{-2}$ times. This result is based on a method we develop here that allows one to determine the elements of all $k$-qubit RDMs, to precision $\epsilon$, by repeating a single quantum circuit for $\lesssim 3^k \epsilon^{-2}$ times, independent of the system size. This improves over existing schemes for determining qubit RDMs.

1 Introduction

The paper motivates simpler, more efficient fermion-to-qubit mappings for quantum simulation and introduces a ternary-tree mapping with low-weight Pauli representations. It also presents a single-circuit approach to learning qubit and fermionic reduced density matrices.

1 Introduction

  • Fermion-to-qubit mapping is an essential step in digital fermionic quantum simulation, including protocols such as the variational quantum eigensolver.Quantum simulation is presented as an alternative approach to hard problems in physics, chemistry, and materials science.Source: §1 · p. 1
  • The ternary-tree mapping represents a single Majorana operator with a Pauli operator acting on ⌈log3(2n + 1)⌉ qubits and is proven optimal in operator weight.For large n, the stated weight is approximately log2 3 ≃1.58 times lower than ⌈log2 n + 1⌉ in the Bravyi-Kitaev transformation.Source: §1 · p. 1
  • ≲(2n + 1)^k/ϵ^2 repetitions suffice to determine all k-fermion RDM elements in parallel to precision ϵ using one quantum circuit.The paper applies the ternary-tree mapping to fermionic RDM learning.Source: §1 · pp. 1–2
  • ≲3^k/ϵ^2 repetitions suffice to determine all k-qubit RDM elements in parallel to precision ϵ, independent of system size n.The method uses Bell-basis measurements and improves the scaling of a recent result by a log n factor.Source: §1 · p. 2
  • The paper develops the mapping and RDM measurement scheme, with further proofs and extensions to qudits in the appendices.The qubit measurement scheme is shown to implement a symmetric informationally complete POVM and is generalized to qudits.Source: §1 · p. 2

2 Fermion-to-qubit mapping

The mapping assigns mutually anticommuting Pauli operators to paths in a ternary tree, using their labels to represent Majorana operators. Its path length sets the operator weight, while the number of paths accommodates the required Majorana operators.

2 Fermion-to-qubit mapping

  • A complete ternary tree associates a qubit with each node except those at the bottom level, and its paths supply candidate operators for the Majorana representation.The construction begins with a complete tree and later treats more general cases.Source: §2 · p. 2
  • Figure 1 illustrates a height-2 tree, labels the qubit associated with each node, and assigns a Pauli operator to every root-to-leaf path.The left-most path corresponds to A0,0 = σx 0σx 1.Source: Figure 1 · p. 2
  • Each root-to-leaf path is specified by a vector p whose entries in {0, 1, 2} select the next branch at each depth.The qubits are indexed consecutively down the tree, and each path is associated with a multi-Pauli operator.Source: §2 · p. 2
  • Pauli operators assigned to distinct paths mutually anticommute because paths diverge and assign different Pauli operators at their first shared qubit of divergence.After divergence, the paths act on different qubits.Source: §2 · p. 2
  • The constructed path operators have weight h = log3(2n + 1), approximately log2 3 ≃1.58 times lower than ⌈log2 n + 1⌉ in the Bravyi-Kitaev transformation.There are 2n + 1 root-to-leaf paths, but the product of all corresponding operators is proportional to identity, leaving 2n independent operators for the Majoranas.Source: §2 · p. 2
  • When 2n + 1 is not a power of 3, the mapping can still be constructed with Pauli weights no greater than ⌈log3(2n + 1)⌉.The passage introduces steps for this non-complete case.Source: §2 · p. 2

1. Introduce a complete ternary tree with height h,

The optimality argument establishes a lower bound on average Pauli weight for any fermion-to-qubit mapping. The ternary-tree construction reaches that bound by balancing the three Pauli operators.

1. Introduce a complete ternary tree with height h,

  • Any fermion-to-qubit mapping has average Pauli weight at least log3(2n), and the ternary-tree mapping is optimal.The proof uses a Gram matrix and expectation values of Majorana operators to establish the bound.Source: §1 · p. 3
  • The proof bounds the squared norm of the vector of Majorana expectation values by 1 for any fermionic state.This follows from the Majorana anticommutation relations.Source: §1 · p. 3
  • The constructed mapping balances the numbers of Pauli-x, -y, and -z operators to attain the lower bound.The proof considers a product state symmetric with respect to σx, σy, and σz, treating equal-weight Pauli expectations on an equal footing.Source: §1 · p. 3

3 Reduced density matrices tomography

The section presents a Bell-basis scheme that estimates all k-qubit RDM elements in parallel, then combines it with the ternary-tree mapping to estimate fermionic k-RDM elements. Both methods use repeated runs of a single circuit.

3.1 Qubits

  • Bell-basis measurements on system–ancilla pairs provide simultaneous measurements of σx ⊗ σx, σy ⊗ σy, and σz ⊗ σz.The Bell basis is a common eigenbasis of these commuting operators, and the measurements can be performed in parallel across qubit pairs.Source: §3.1 · pp. 3–4, Table 1 · p. 4
  • ≲3^k/ϵ^2 repetitions of one circuit determine all k-qubit RDM elements in parallel to precision ϵ, independent of system size n.The scheme measures all k-RDM elements at once.Source: §3.1 · p. 4
  • The Bell-basis scheme is informationally complete, allowing reconstruction of the entire n-qubit state rather than only RDMs up to a specified order.For finite measurement rounds, smaller-k RDMs are estimated more accurately than larger-k RDMs.Source: §3.1 · p. 4

3.2 Fermions

  • Ternary-tree encoding makes fermionic k-RDM elements involve Pauli operators of weight at most 2k log3(2n+1).This mapping-based bound is used with the Bell-basis measurement strategy to obtain the fermionic repetition bound.Source: §3.2 · p. 4
  • ≲(2n + 1)^k/ϵ^2 repetitions of one circuit determine all fermionic k-RDM elements to precision ϵ.The bound follows from combining the ternary-tree mapping with Bell-basis measurement.Source: §3.2 · p. 4

4 Conclusion

The conclusion presents a ternary-tree mapping that saturates a lower bound on Pauli weights and uses it to measure fermionic RDM elements with one repeated circuit. It also identifies future directions for varying tree depths and combining the mapping with error-correcting codes.

4 Conclusion

  • The ternary-tree mapping saturates the lower bound on Pauli weights for single fermionic operators, balancing the numbers of Pauli-x, -y, and -z operators.The authors expect it to be useful for studying fermionic systems on noisy and fault-tolerant quantum computing devices.Source: §4 · p. 5
  • ≲(2n + 1)^k/ϵ^2 repetitions of one circuit determine fermionic k-RDM elements to precision ϵ; the Bravyi-Kitaev transformation has a worse repetition exponent.The fermionic operators’ noncommuting feature prevents simultaneous measurement.Source: §4 · p. 5
  • The Bell-basis method estimates all k-qubit RDM elements in parallel with ≲3^kϵ^-2 repetitions, independent of system size, and is informationally complete.The scheme uses extra ancilla qubits compared with the cited Cotler and Wilczek results, and the paper also generalizes it to qudits using Heisenberg-Weyl operators.Source: §4 · p. 5
  • Future work includes varying ternary-tree path depths to prioritize important Majorana operators and combining the mapping with quantum error-correcting codes using redundant qubits.Source: §4 · p. 5

A The informational completeness of the Bell-basis measurement and its implementation

The appendix proves that the Bell-basis measurement scheme implements a qubit SIC POVM and gives a condition for informational completeness. It also relates the construction to dense coding and Heisenberg–Weyl covariant SIC POVMs.

A The informational completeness of the Bell-basis measurement and its implementation

  • The Bell-basis measurement scheme implements a symmetric informationally complete POVM on a qubit.The appendix derives the effective POVM from an ancilla-assisted Bell-basis measurement and verifies the SIC condition for its four outcomes.Source: A The informational completeness of… · pp. 7–8
  • If Tr(σαξ) ≠ 0 for α = x, y, z, the procedure is informationally complete for ρ in the infinite-repetition limit.Under this condition, the measurement outcomes suffice to reconstruct ρ.Source: A The informational completeness of… · p. 7
  • The Bell basis is a common eigenbasis of σx⊗σx, σy⊗σy, and σz⊗σz, so these pairwise observables can be read from its outcomes.The appendix uses this basis property in discussing the resulting correlation functions.Source: A The informational completeness of… · p. 7
  • The four SIC POVM elements are defined as Ei = 1/2|ψi⟩⟨ψi| for i = 1, . . . , 4.The appendix checks their SIC property using the ancilla state and its Pauli conjugates.Source: A The informational completeness of… · pp. 7–8
  • The Bell-measurement construction connects dense coding to Heisenberg–Weyl group-covariant SIC POVMs; choosing ξ as the fiducial state recovers the tetrahedron measurement.The appendix identifies the relations from dense coding as the structure underlying the POVM equations.Source: A The informational completeness of… · p. 8

B Generalization of RDM measurement scheme to qudits

The paper generalizes its Bell-basis RDM measurement scheme to qudits using Heisenberg–Weyl operators and a generalized Bell basis. The scheme estimates all k-qudit RDM elements with a repetition count independent of system size, and is informationally complete under a stated condition.

B Generalization of RDM measurement scheme to qudits

  • Measuring the required Heisenberg–Weyl correlation functions with one quantum circuit determines all k-qudit RDMs.The scheme attaches a known-state ancilla to each system qudit and measures each pair in the generalized Bell basis.Source: B Generalization of RDM measurement… · pp. 8–9
  • The generalized Bell basis is an orthonormal two-qudit basis in which the operators X^fZ^g ⊗ X^fZ^-g are jointly measurable.Their mutual commutation means the basis measurement reveals their values simultaneously.Source: B Generalization of RDM measurement… · p. 9
  • (D + 1)^k/ϵ^2 repetitions suffice to estimate elements of all k-qudit RDMs to error ϵ, independently of n.The stated repetition count gives fixed standard-deviation error and uses a single measurement procedure.Source: B Generalization of RDM measurement… · p. 9
  • If Tr(X^fZ^-gξ) ≠ 0 for all f, g = 0, . . . , D − 1, the procedure is informationally complete for ρ in the infinite-repetition limit.The generalized Bell basis permits recovery of the correlation functions used to establish this result.Source: B Generalization of RDM measurement… · p. 9
  • The measurement scheme implements a qudit SIC POVM when ξ is a fiducial state, but finding such states remains open in some dimensions.The text says such states are known in various dimensions and believed to exist in every dimension; for selected correlations, a less stringent condition can be used.Source: B Generalization of RDM measurement… · pp. 9–10

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