[Paper Review] Generalised Clifford groups and simulation of associated quantum circuits

Sean Clark, Richard Jozsa+1 more|ArXiv.org|Jan 15, 2007

Quantum Computing Algorithms and Architecture13 references4 citations

TL;DR

This paper generalizes the Gottesman-Knill theorem by replacing the Pauli group with arbitrary finite subgroups $ G \subset U(d) $, identifying conditions under which the projective normalizer of $ G \otimes G $ yields entangling gates. For qubits ($ d=2 $), it exhaustively classifies such groups and finds that only those generated by $ X $ and the $ n $th root of $ Z $ admit entangling normalizers, extending the scope of classically simulable quantum circuits beyond the standard Clifford group.

ABSTRACT

Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G tensor G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2 we exhaustively treat all finite subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups G_n generated by X and the n^th root of Z.

Motivation & Objective

• To generalize the Gottesman-Knill theorem by replacing the Pauli group with arbitrary finite subgroups $ G \subset U(d) $, aiming to identify new classes of quantum circuits that are classically efficiently simulable.
• To investigate when the projective normalizer of $ G \otimes G $ is entangling, as this leads to simulable quantum circuits via a generalized Gottesman-Knill framework.
• To exhaustively classify all finite subgroups of $ U(2) $ up to unitary equivalence and phase extensions that yield entangling normalizers of $ G \otimes G $, focusing on the qubit case.
• To explore whether higher-dimensional groups ($ d \geq 3 $) or reducible representations could yield new entangling normalizers beyond the qubit case.
• To identify mathematical signatures of groups $ G $ that ensure the existence of non-trivial projective normalizers, particularly in relation to circuit composition and simulation.

Proposed method

• Define generalized Clifford groups as the projective normalizers of $ G \otimes G $, where $ G $ is a finite subgroup of $ U(d) $, and study their structure for entangling properties.
• Use group-theoretic techniques to analyze the normalizer of $ G \otimes G $ in $ U(2^2) $, focusing on the projective normalizer structure and its relation to entanglement generation.
• Apply computational algebra procedures to classify all finite subgroups of $ U(2) $, up to unitary equivalence and phase extensions, by analyzing their normalizers and identifying those with entangling normalizers.
• Characterize the normalizer structure of $ G_m = \langle X, Z^{1/m} \rangle $, showing that $ CZ $ and SWAP generate the full normalizer of $ G_m \otimes G_m $, and that $ CZ $ is an entangling gate for all $ m \in \mathbb{N} $.
• Use representation theory of finite groups, particularly the binary dihedral group $ Q_{4n} $, to prove that no new projective normalizer structures arise beyond those isomorphic to $ G_m $ for $ m \in \mathbb{N} $.
• Investigate the structural properties of normalizers of $ G^{igotimes n} $ for $ n \geq 3 $, asking whether new normalizers exist beyond those composed from $ n=1 $ and $ n=2 $ components.

Experimental results

Research questions

• RQ1 Which finite subgroups $ G \subset U(2) $ have projective normalizers of $ G \otimes G $ that are entangling, leading to classically simulable quantum circuits?
• RQ2 Are there finite subgroups $ G \subset U(2) $, beyond the standard Pauli group and its extensions, that yield new classes of efficiently simulable quantum circuits via generalized normalizer structures?
• RQ3 Can the normalizer of $ G^{igotimes n} $ for $ n \geq 3 $ contain gates not generated by $ n=1 $ and $ n=2 $ normalizers, even if $ G \otimes G $ has no such entangling normalizers?
• RQ4 What mathematical properties of a finite group $ G \subset U(d) $ signal the existence of non-trivial projective normalizers that generate entangling operations?
• RQ5 Do higher-dimensional groups ($ d \geq 3 $) or reducible representations of $ G \subset U(d) $ yield new entangling normalizers not present in the irreducible case?

Key findings

• For qubits ($ d=2 $), the only finite subgroups $ G \subset U(2) $ (up to unitary equivalence and phase extensions) with entangling normalizers of $ G \otimes G $ are those generated by the Pauli $ X $ gate and the $ n $th root of the $ Z $ gate, denoted $ G_m $ for $ m \in \mathbb{N} $.
• The projective normalizer of $ G_m \otimes G_m $ is generated by the controlled-$ Z $ gate $ CZ $, the swap gate, and $ G_m \otimes G_m $, and $ CZ $ acts as an entangling gate for all $ m \in \mathbb{N} $.
• All finite subgroups of $ U(2) $ with central quotient isomorphic to the dihedral group $ D_{2n} $ (for $ n $ even) are unitarily equivalent to central extensions of $ G_m $, and no new normalizer structures arise for different irreducible representations of the binary dihedral group $ Q_{4n} $.
• The normalizer structure of $ G_m \otimes G_m $ is fully characterized and shown to be isomorphic to the standard Clifford group structure, with no new entangling normalizers beyond those already known from the Pauli group.
• For $ d \geq 3 $, computational searches on $ U(3) $ did not yield new entangling teleportation groups, though exhaustive classification remains computationally challenging due to group size.
• The paper leaves open whether $ G^{igotimes n} $ for $ n \geq 3 $ can have normalizers not composed of $ n=1 $ and $ n=2 $ components, even if $ G \otimes G $ has no such entangling normalizers.