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arXiv:1904.09912 · Quanta

Clifford Algebras, Spin Groups and Qubit Trees

Alexander Yu. Vlasov

Published Dec 1, 2022• 15 citations
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Abstract

Representations of Spin groups and Clifford algebras derived from the structure of qubit trees are introduced in this work. For ternary trees the construction is more general and reduction to binary trees is formally defined by deletion of superfluous branches. The usual Jordan–Wigner construction also may be formally obtained in this approach by bringing the process up to trivial qubit chain (trunk). The methods can also be used for effective simulation of some quantum circuits corresponding to the binary tree structure. The modeling of more general qubit trees, as well as the relationship with the mapping used in the Bravyi–Kitaev transformation, are also briefly discussed. Quanta 2022; 11: 97–114.

DOI
10.12743/quanta.v11i1.199

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Representations of Spin groups and Clifford algebras derived from the structure of qubit trees are introduced in this work. For ternary trees the construction is more general and reduction to binary trees is formally defined by deletion of superfluous branches. The usual Jordan–Wigner construction also may be formally obtained in this approach by bringing the process up to trivial qubit chain (trunk). The methods can also be used for effective simulation of some quantum circuits corresponding to the binary tree structure. The modeling of more general qubit trees, as well as the relationship with the mapping used in the Bravyi–Kitaev transformation, are also briefly discussed. Quanta 2022; 11: 97–114.
Algebras with given (anti-)commutativity structure are widespread in quantum mechanics. This structure is captured by quasi-Clifford algebras (QCA): a QCA generated by α1,…,αnα_1, \dots, α_nα1​,…,αn​ is is given by the relations αi2=kiα_i^2 = k_iαi2​=ki​ and αjαi=(−1)χijαiαjα_j α_i = (-1)^{χ_{ij}} α_i α_jαj​αi​=(−1)χij​αi​αj​, where ki∈Ck_i \in \mathbb{C}ki​∈C and χij∈{0,1}χ_{ij} \in \{0, 1\}χij​∈{0,1}. We present a mapping from QCA to Pauli algebras and discuss its use in quantum information and computation. The mapping also provides a Wedderburn decomposition of matrix groups with quasi-Clifford structure. This provides a block-diagonalization for e.g. Pauli groups, while for Majorana operators the Jordan-Wigner transform is recovered. Applications to the symmetry reduction of semidefinite programs and for constructing maximal anti-commuting subsets are discussed.
We consider two approaches to designing fermion-qubit mappings: (1) ternary tree transformations, which use Pauli representations of the Majorana operators that correspond to root-to-leaf paths of a tree graph and (2) linear encodings of the Fock basis, such as the Jordan-Wigner and Bravyi-Kitaev transformations, which store linear binary transformations of the fermionic occupation number vectors in the computational basis of qubits. These approaches have emerged as distinct concepts, with little notational consistency between them. In this paper we propose a universal description of fermion-qubit mappings, which reveals the relationship between ternary tree transformations and linear encodings. Using our notation, we show that every product-preserving ternary tree transformation is equivalent to a linear encoding of the Fock basis.
Standard fermion-to-qubit encodings (Jordan–Wigner, Bravyi–Kitaev, the Bravyi–Kitaev tree, parity, ternary tree) are related by Clifford transformations, so they form a gauge group acting on the classical-simulability problem. This code demonstrates the central split: the Pauli weight and causal-cone width are gauge-covariant (re-encoding can lower them exponentially), while the dynamical-Lie-algebra dimension and the non-stabilizerness (magic) are gauge-invariant. Trainability — the cost landscape, the gradient, and the barren-plateau variance — is gauge-invariant to machine precision.
In order to simulate a system of fermions on a quantum computer, it is necessary to represent the fermionic states and operators on qubits. This can be accomplished in multiple ways, including the well-known Jordan-Wigner transform, as well as the parity, Bravyi-Kitaev, and ternary tree encodings. Notably, the Bravyi-Kitaev encoding can be described in terms of a classical data structure, the Fenwick tree. Here we establish a correspondence between a class of classical data structures similar to the Fenwick tree, and a class of one-to-one fermion-to-qubit transforms. We present a novel fermion-to-qubit encoding based on the recently discovered "Sierpinski tree" data structure, which matches the operator locality of the ternary tree encoding, and has the additional benefit of encoding the fermionic states as computational basis states. This is analogous to the formulation of the Bravyi-Kitaev encoding in terms of the Fenwick tree.