Alison Warman and Sakura Schäfer-Nameki of the Mathematical Institute, University of Oxford, have devised a method for creating topologically protected phase gates in two dimensions, bypassing limitations imposed by the Bravyi-König theorem. Their work encodes a logical qubit using the quantum double of the dihedral group D_(4N) on a triangular spatial patch, enabling constant-depth unitary gates at any level of the Clifford hierarchy.

Specifically, the researchers realize the phase gate T^(1/N) = diag(1,e^(iπ/(4N))) in the logical Z basis, and demonstrate that for 8N = 2^n, the logical gate lies at the n-th level of the Clifford hierarchy and has a qubit-only realization; it can be constructed in terms of Clifford-hierarchy stabilizers for a code with n physical qubits on each edge of the lattice.

D(4N) Groups Enable Constant-Depth Clifford-Hierarchy Gates

This construction bypasses a limitation known as the Bravyi-König theorem, which previously restricted constant-depth quantum circuits on Pauli stabilizer codes to the D-th level of the hierarchy, and opens new avenues for building more efficient quantum computers. This approach leverages non-Abelian surface codes, expanding the range of available logicals and permitting diagonal gates at any Clifford-hierarchy level without requiring non-local operations or layouts extending beyond two dimensions.

This represents a significant reduction in qubit overhead compared to traditional approaches, potentially accelerating the development of scalable quantum processors. The paper states, “Constant-Depth Non-Clifford Gates from D(D_(4N)),” and continues, “For any integer N ≥ 1, consider the order-8N dihedral group.” The team’s approach proposes a non-abelian stabilizer group formalism, which they develop for dihedral groups.

2D Non-Abelian Surface Codes Bypass Bravyi-König Theorem

This new approach utilizes non-Abelian surface codes, specifically leveraging the dihedral group D_(4N) to realize phase gates, which are essential for universal quantum computation, with a constant-depth circuit. The work offers a pathway toward more efficient quantum computers by potentially reducing the qubit overhead typically required for error correction. Unlike Pauli stabilizer codes, which are subject to the constraints of the Bravyi-König theorem, these codes originate from non-Abelian topological order, meaning the stabilizer group is not necessarily Abelian.

This distinction is critical, as it allows for the construction of unitary gates at arbitrary levels of the Clifford hierarchy, a measure of gate complexity, without increasing the circuit depth. The researchers detail that the logical gate is implemented by stacking a symmetry-protected topological phase, specified by a group 2-cocycle and boundary counter-terms, onto the spatial region of the code.

This construction bypasses the theorem’s limitations by operating in a different mathematical space. The researchers state, “We will clarify this in the main text of the paper, but let us briefly explain the setting here,” emphasizing the foundational nature of their work. To achieve a universal gate set, the team proposes switching to the double surface code, enabling the generation of all necessary Clifford gates within the system.

They substantiate their findings with a concrete lattice description and prove Theorem 1, detailing how to construct unitary operators that implement automorphisms of D(G) and are topologically protected. They further demonstrate, through Corollary 1, the realization of constant-depth T^(1/N) gates, essential components of the single qubit Clifford hierarchy.

The logical states formed by this method constitute a complete qubit logical basis for the non-abelian code D(G) on a triangle, functioning as the +1 and -1 eigenstates of the logical Z operator. This alternative to the Bravyi-König theorem, by focusing on purely 2D codes with increased complexity through non-Abelian surface codes, is expected to provide an advantage in implementation.

The physical gates required for this approach, however, are non-Clifford, presenting a new set of challenges for hardware development. The researchers suggest that further exploration of a just-in-time decoder geared towards non-Abelian surface codes will be crucial for realizing the full potential of this method.

Dihedral Group D_(4N) Realizes T^(1/N) Phase Gate

Researchers are constructing quantum gates using the dihedral group D_(4N), a mathematical structure that allows for constant-depth circuits in two dimensions and bypasses limitations found in traditional quantum computing approaches. The paper details, “A logical state is specified by anyons forming a trivalent junction,” explaining how the red and blue anyons fuse to form an orange anyon that can terminate on a boundary. In this context, they propose a non-abelian stabilizer group formalism, which they develop for dihedral groups, to define logical states and implement the T^(1/N) phase gate through a constant-depth circuit.

They also discuss code-switching to the double surface-code to complete a universal gate-set in this setup. The team’s work demonstrates a family of 2D codes where a single logical qubit can implement the phase gate with constant depth, a crucial step toward scalable and universal quantum computation.

SPT Phase Construction with 2-Cocycles & Boundary Terms

The construction of robust quantum gates hinges on minimizing the physical resources required, and a new approach leverages the dihedral group D_(4N) to achieve constant-depth phase gates with a reduced qubit overhead. The team’s setup utilizes a prism-like space-time geometry, with triangular spatial cross-sections and vertical faces defining topological boundary conditions. This arrangement allows for the manipulation of anyons, quasiparticles exhibiting exotic exchange statistics, to encode and process quantum information. The researchers detail how the operators necessary for gate construction can be realized with (n-1)-qubit gates, streamlining the circuit complexity.

For example, the C gate is mapped to a series of X and CX gates acting on qubits, while the Z gate incorporates phase gates with roots of unity determined by supersolvable generators. The paper explains, “Each vertex term will multiply two adjacent boundary edges by g and g^(-1) respectively,” detailing how the boundary terms contribute to the overall gate operation. Careful selection of boundary conditions and the application of a mapping, M^β, ensures that all states in the superposition carry the same phase, simplifying the evaluation of the gate.

The algebra A(K,N) provides a map between the anyons of the two topological orders, creating a gapped boundary condition of the folded quantum double D(G× K/N). This ability to transition between different codes expands the computational possibilities within this framework, moving beyond solely phase-based operations and towards more complex quantum algorithms.

Non-Abelian Stabilizer Formalism for Dihedral Groups

This qubit-only realization is achieved by constructing the gate using Clifford-hierarchy stabilizers for a code with n physical qubits on each edge of the lattice. Previous explorations of non-abelian stabilizer groups have focused on qubit and permutation group stabilizers, but this work proposes a non-abelian stabilizer group formalism, which they develop for dihedral groups. The researchers’ approach defines a non-Abelian stabilizer code by establishing suitable boundary conditions for a surface code on a triangular patch, realizing a single logical qubit, and then determining the complete stabilizer group, which does not fully commute.

The local Hilbert space on each edge of the lattice is spanned by group elements, utilizing a Z_4-qudit and a qubit, and the logical subspace is characterized by the inequivalent ground states of the quantum double Hamiltonian with boundaries.

The paper explains, “We first recover, in this SPT-stacking language, the familiar constant-depth T = P(π/4) in D(D_4), a non-Abelian presentation equivalent to D^ω(Z_2^3), and then exhibit a systematic generalization to the family of dihedral groups D_(4N) where the same mechanism yields T^(1/N).” Beyond phase gates, this framework allows for code-switching to the double surface-code, completing a universal gate-set and expanding computational possibilities.

Clifford Hierarchy Level & Physical Qubit Count for 8N=2^n

The ability to construct a logical gate at a specific level of the Clifford hierarchy, using a physical Hilbert space with a limited number of qubits per lattice edge, represents an advance in fault-tolerant quantum computation. This contrasts with traditional approaches that often require a substantially larger number of physical qubits to encode a single logical qubit and perform complex operations. The team’s work bypasses limitations imposed by the Bravyi-König theorem, which restricts the complexity of gates implementable in constant depth on Pauli stabilizer codes in lower dimensions.

The resulting unitary, U_α,β, acts on the logical qubit as, effectively implementing a phase gate at the n-th level. The researchers state, “In summary we obtain a constant-depth topologically protected gate at level n of the Clifford hierarchy.”

A table within the paper details the resources required to realize the non-Clifford gate T^1/N for various values of n, illustrating the scalability of the approach. Values of N beyond this point would necessitate higher-order qudits for implementation. This qubit-only realization is crucial, as it simplifies the physical architecture required for building a scalable quantum computer.

👉 More information
🗞 Constant-Depth Clifford-Hierarchy Gates via Non-Abelian Surface Codes
✍️ Alison Warman and Sakura Schäfer-Nameki
🧠 DOI: http://link.aps.org/doi/10.1103/kfxl-rv7c

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