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arXiv:2410.01887

Matchgate hierarchy: A Clifford-like hierarchy for deterministic gate teleportation in matchgate circuits

Angelos Bampounis, Rui Soares Barbosa, Nadish de Silva

Published Oct 2, 2024• 0 citations
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Abstract

The Clifford hierarchy, introduced by Gottesman and Chuang in 1999, is an increasing sequence of sets of quantum gates crucial to the gate teleportation model for fault-tolerant quantum computation. Gates in the hierarchy can be deterministically implemented, with increasing complexity, via gate teleportation using (adaptive) Clifford circuits with access to magic states. We propose an analogous gate teleportation protocol and a related hierarchy in the context of matchgate circuits, another restricted class of quantum circuits that can be efficiently classically simulated but are promoted to quantum universality via access to `matchgate-magic' states. The protocol deterministically implements any $n$-qubit gate in the hierarchy using adaptive matchgate circuits with magic states, with the level in the hierarchy indicating the required depth of adaptivity and thus number of magic states consumed. It also provides a whole family of novel deterministic matchgate-magic states. We completely characterise the gates in the matchgate hierarchy for two qubits, with the consequence that, in this case, the required number of resource states grows linearly with the target gate's level in the hierarchy. For an arbitrary number of qubits, we propose a characterisation of the matchgate hierarchy by leveraging the fermionic Stone$\unicode{x2013}$von Neumann theorem. It places a polynomial upper bound on the space requirements for representing gates at each level.

DOI
10.48550/arxiv.2410.01887

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The Clifford hierarchy, introduced by Gottesman and Chuang in 1999, is an increasing sequence of sets of quantum gates crucial to the gate teleportation model for fault-tolerant quantum computation. Gates in the hierarchy can be deterministically implemented, with increasing complexity, via gate teleportation using (adaptive) Clifford circuits with access to magic states. We propose an analogous gate teleportation protocol and a related hierarchy in the context of matchgate circuits, another restricted class of quantum circuits that can be efficiently classically simulated but are promoted to quantum universality via access to `matchgate-magic' states. The protocol deterministically implements any nnn-qubit gate in the hierarchy using adaptive matchgate circuits with magic states, with the level in the hierarchy indicating the required depth of adaptivity and thus number of magic states consumed. It also provides a whole family of novel deterministic matchgate-magic states. We completely characterise the gates in the matchgate hierarchy for two qubits, with the consequence that, in this case, the required number of resource states grows linearly with the target gate's level in the hierarchy. For an arbitrary number of qubits, we propose a characterisation of the matchgate hierarchy by leveraging the fermionic Stone\unicodex2013\unicode{x2013}\unicodex2013von Neumann theorem. It places a polynomial upper bound on the space requirements for representing gates at each level.
Matchgate unitaries are ubiquitous in quantum computation due to their relation to non-interacting fermions and because they can be used to benchmark quantum computers. Implementing such unitaries on fault-tolerant devices requires first compiling them into a discrete universal gate set, typically Clifford+T+T+T. Here, we propose a different approach for their synthesis: compile matchgate unitaries using only matchgate gates. To this end, we first show that the matchgate-Clifford group (the intersection of the matchgate and Clifford groups) plus the T‾\overline{T}T gate (a TTT unitary up to a phase) is universal for the matchgate group. Our approach leverages the connection between nnn-qubit matchgate circuits and the standard representation of SO(2n)\mathbb{SO}(2n)SO(2n), which reduces the compilation from 2n×2n2^n\times 2^n2n×2n unitaries to 2n×2n2n\times2n2n×2n ones, thus reducing exponentially the size of the target matrix. Moreover, we rigorously show that this scheme is efficient, as an approximation error εSO(2n)\varepsilon_{\mathbb{SO}(2n)}εSO(2n)​ incurred in this smaller-dimensional representation translates at most into an O(n εSO(2n))O(n \,\varepsilon_{\mathbb{SO}(2n)})O(nεSO(2n)​) error in the exponentially large unitary. In addition, we study the exact version of the matchgate synthesis problem, and we prove that all matchgate unitaries UUU such that U⊗U∗U\otimes U^*U⊗U∗ has entries in the ring Z[1/2,i]\mathbb{Z}\big[1/\sqrt 2,i\big]Z[1/2​,i] can be exactly synthesized by a finite sequence of gates from the matchgate-Clifford+T‾+\overline{T}+T set, without ancillas. We then use this insight to map optimal exact matchgate synthesis to Boolean satisfiability, and compile the circuits that diagonalize the free-fermionic XXXXXX Hamiltonian on n=4, 8n=4,\,8n=4,8 qubits.
The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The groups of Pauli and Clifford gates constitute the first and second ‘levels’, respectively. Non-Clifford gates from the third level or higher, such as the T gate, are necessary for achieving fault-tolerant universal quantum computation. Since it was defined 26 years ago by Gottesman–Chuang, two questions have been studied by numerous researchers. First, precisely which gates constitute the Clifford hierarchy? Second, which subset of the hierarchy gates admit efficient gate teleportation protocols? We completely solve both questions in the practically relevant case of the Clifford hierarchy for gates of one qubit or one qudit of prime dimension. We express every such hierarchy gate uniquely as a product of three simple gates, yielding also a formula for the size of every level. These results are a consequence of our finding that all such hierarchy gates can be expressed in a certain form that guarantees efficient gate teleportation. Our decomposition of Clifford gates as a unique product of three elementary Clifford gates is of broad applicability.
Quantum gate teleportation is a key technique in fault-tolerant quantum computation that uses resource states to implement logical gates. Here, we develop a theory of magic gate teleportation (MGT), a form of quantum gate teleportation that uses non-stabilizer resource states to implement non-Clifford unitary gates on arbitrary input states without revealing information about those states. This setting is motivated by the canonical approach to implementing logical non-Clifford gates in fault-tolerant quantum computation. We uncover a hidden structure within MGT---after backpropagating the Pauli measurements, MGT protocols can be viewed as encoding the input state into a stabilizer code heralded by the measurement outcomes, followed by a logical non-Clifford gate. Using this structure, we construct MGT protocols for any resource state obtained by applying commuting Pauli rotations to a stabilizer state, and provide an efficient algorithm for synthesizing their circuit implementations. Conversely, we prove that directly useful single-qubit resource states are necessarily Clifford-equivalent to diagonal states, and establish the same constraint for multi-qubit resource states used in a broad class of MGT protocols. In particular, the output state distilled from the [ ⁣[5,1,3] ⁣][\![5, 1, 3]\!][[5,1,3]] protocol is not directly useful for any MGT protocols, though it can be probabilistically converted into directly useful states. Finally, we identify conditions under which the feedforward operators can be implemented by Pauli operators, shedding light on the paradigm of algorithmic fault tolerance and simplifying the feedforward operations needed for quantum computing.
Clifford and matchgate circuits are canonical families of classically simulable quantum circuits. Their intersection, the matchgate-Clifford group, plays an important role in randomized fermionic protocols and in matchgate synthesis. Its adjoint action is isomorphic to the group of unit-determinant signed permutations of 2n2n2n Majorana modes, and we study optimal exact synthesis in this group. That is, given a target unitary and a gate set, output an nnn-qubit circuit implementing the target using the fewest operations. We show that the complexity of this problem strongly depends on the gate set. In particular, we study gate sets consisting of Majorana braids with different connectivity graphs. For path and complete graphs, we prove that the problem is classically solvable in O(n2)\mathcal{O}\left(n^2\right)O(n2) time, and we provide explicit gate-optimal compilers. In addition, we prove that when the connectivity graph is a tree, the decision version of the optimal synthesis problem becomes NP-complete. Finally, we benchmark our optimal compiler on chains of up to n=80n=80n=80 qubits against those of \texttt{Qiskit} and \texttt{Tket}, obtaining circuits with constant factor improvements ×2.57\times2.57×2.57 and ×2.28\times2.28×2.28 in the total number of gates, respectively.
The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The importance of these gates has motivated numerous studies of their structure. Zeng-Chen-Chuang conjectured in 2007 that all hierarchy gates are generalised semi-Clifford, i.e. take the form C1ΠDC2C_1 ΠD C_2C1​ΠDC2​ for Clifford gates C1,C2C_1, C_2C1​,C2​, a permutation gate ΠΠΠ, and a diagonal gate DDD; Beigi-Shor proved in 2008 that this holds for all third-level gates. We construct a five-qubit gate that is in the fifth level of the Clifford hierarchy but is not generalised semi-Clifford. Rather than simply present and verify our counterexample to the generalised semi-Clifford conjecture, we show how its form can be deduced. Our counterexample also demonstrates that the Clifford hierarchy is not closed under inverses.
We develop a simple and systematic route to higher levels of the qubit Clifford hierarchy by coherently controlling Clifford operations. Our approach is based on Pauli periodicity, defined for a Clifford unitary UUU as the smallest integer m≥1m\ge 1m≥1 such that U2mU^{2^{m}}U2m is a Pauli operator up to phase. We prove a sharp controlled-jump rule showing that the controlled gate CUCUCU lies strictly in level m+2m+2m+2 of the hierarchy, and equivalently that CUCUCU lies in level kkk if U2k−2U^{2^{k-2}}U2k−2 is Pauli while no smaller positive power of UUU is Pauli. We further quantify the resources required to realize large level jumps in the Clifford hierarchy by proving an essentially tight upper bound on Pauli periodicity as a function of the number of qubits, which implies that accessing high hierarchy levels through controlled Cliffords requires a number of target qubits that grows exponentially with the desired level. We complement this limitation with explicit infinite families of Pauli-periodic Cliffords whose controlled versions achieve asymptotically optimal jumps. As an application, we propose a protocol for preparing logical catalyst states that enable logical Z1/2kZ^{1/2^k}Z1/2k phase gates via phase kickback from a single jumped Clifford.