Clifford Hierarchy Research Articles

On Groups in the Qubit Clifford Hierarchy

On Groups in the Qubit Clifford Hierarchy

Quantum advantage in temporally flat measurement-based quantum computation

Several classes of quantum circuits have been shown to provide a quantum computational advantage under certain assumptions. The study of ever more restricted classes of quantum circuits capable of quantum advantage is motivated by possible simplifications in experimental demonstrations. In this paper we study the efficiency of measurement-based quantum computation with a completely flat temporal ordering of measurements. We propose new constructions for the deterministic computation of arbitrary Boolean functions, drawing on correlations present in multi-qubit Greenberger, Horne, and Zeilinger (GHZ) states. We characterize the necessary measurement complexity using the Clifford hierarchy, and also generally decrease the number of qubits needed with respect to previous constructions. In particular, we identify a family of Boolean functions for which deterministic evaluation using non-adaptive MBQC is possible, featuring quantum advantage in width and number of gates with respect to classical circuits.

Transversal diagonal logical operators for stabiliser codes

Storing quantum information in a quantum error correction code can protect it from errors, but the ability to transform the stored quantum information in a fault tolerant way is equally important. Logical Pauli group operators can be implemented on Calderbank-Shor-Steane (CSS) codes, a commonly-studied category of codes, by applying a series of physical Pauli X and Z gates. Logical operators of this form are fault-tolerant because each qubit is acted upon by at most one gate, limiting the spread of errors, and are referred to as transversal logical operators. Identifying transversal logical operators outside the Pauli group is less well understood. Pauli operators are the first level of the Clifford hierarchy which is deeply connected to fault-tolerance and universality. In this work, we study transversal logical operators composed of single- and multi-qubit diagonal Clifford hierarchy gates. We demonstrate algorithms for identifying all transversal diagonal logical operators on a CSS code that are more general or have lower computational complexity than previous methods. We also show a method for constructing CSS codes that have a desired diagonal logical Clifford hierarchy operator implemented using single qubit phase gates. Our methods rely on representing operators composed of diagonal Clifford hierarchy gates as diagonal XP operators and this technique may have broader applications.

Hierarchies of resources for measurement-based quantum computation

For certain restricted computational tasks, quantum mechanics provides a provable advantage over any possible classical implementation. Several of these results have been proven using the framework of measurement-based quantum computation (MBQC), where nonlocality and more generally contextuality have been identified as necessary resources for certain quantum computations. Here, we consider the computational power of MBQC in more detail by refining its resource requirements, both on the allowed operations and the number of accessible qubits. More precisely, we identify which Boolean functions can be computed in non-adaptive MBQC, with local operations contained within a finite level in the Clifford hierarchy. Moreover, for non-adaptive MBQC restricted to certain subtheories such as stabiliser MBQC, we compute the minimal number of qubits required to compute a given Boolean function. Our results point towards hierarchies of resources that more sharply characterise the power of MBQC beyond the binary of contextuality vs non-contextuality.

Topological Order, Quantum Codes, and Quantum Computation on Fractal Geometries

We investigate topological order on fractal geometries embedded in $n$ dimensions. In particular, we diagnose the existence of the topological order through the lens of quantum information and geometry, i.e., via its equivalence to a quantum error-correcting code with a macroscopic code distance or the presence of macroscopic systoles in systolic geometry. We first prove a no-go theorem that $\mathbb{Z}_N$ topological order cannot survive on any fractal embedded in 2D. For fractal lattice models embedded in 3D or higher spatial dimensions, $\mathbb{Z}_N$ topological order survives if the boundaries of the interior holes condense only loop or membrane excitations. Moreover, for a class of models containing only loop or membrane excitations, and are hence self-correcting on an $n$-dimensional manifold, we prove that topological order survives on a large class of fractal geometries independent of the type of hole boundaries. We further construct fault-tolerant logical gates using their connection to global and higher-form topological symmetries. In particular, we have discovered a logical CCZ gate corresponding to a global symmetry in a class of fractal codes embedded in 3D with Hausdorff dimension asymptotically approaching $D_H=2+\epsilon$ for arbitrarily small $\epsilon$, which hence only requires a space-overhead $\Omega(d^{2+\epsilon})$ with $d$ being the code distance. This in turn leads to the surprising discovery of certain exotic gapped boundaries that only condense the combination of loop excitations and gapped domain walls. We further obtain logical $\text{C}^{p}\text{Z}$ gates with $p\le n-1$ on fractal codes embedded in $n$D. In particular, for the logical $\text{C}^{n-1}\text{Z}$ in the $n^\text{th}$ level of Clifford hierarchy, we can reduce the space overhead to $\Omega(d^{n-1+\epsilon})$. Mathematically, our findings correspond to macroscopic relative systoles in fractals.

Finding the disjointness of stabilizer codes is NP-complete

The disjointness of a stabilizer code is a quantity used to constrain the level of the logical Clifford hierarchy attainable by transversal gates and constant-depth quantum circuits. We show that for any positive integer constant $c$, the problem of calculating the $c$-disjointness, or even approximating it to within a constant multiplicative factor, is NP-complete. We provide bounds on the disjointness for various code families, including the Calderbank-Shor-Steane codes, concatenated codes, and hypergraph product codes. We also describe numerical methods of finding the disjointness, which can be readily used to rule out the existence of any transversal gate implementing some non-Clifford logical operation in small stabilizer codes. Our results indicate that finding fault-tolerant logical gates for generic quantum error-correcting codes is a computationally challenging task.

Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted

We evaluate the usefulness of holographic stabilizer codes for practical purposes by studying their allowed sets of fault-tolerantly implementable gates. We treat them as subsystem codes and show that the set of transversally implementable logical operations is contained in the Clifford group for sufficiently localized logical subsystems. As well as proving this concretely for several specific codes, we argue that this restriction naturally arises in any stabilizer subsystem code that comes close to capturing certain properties of holography. We extend these results to approximate encodings, locality-preserving gates, certain codes whose logical algebras have non-trivial centers, and discuss cases where restrictions can be made to other levels of the Clifford hierarchy. A few auxiliary results may also be of interest, including a general definition of entanglement wedge map for any subsystem code, and a thorough classification of different correctability properties for regions in a subsystem code.

Efficient quantum gate teleportation in higher dimensions

The Clifford hierarchy is a nested sequence of sets of quantum gates critical to achieving fault-tolerant quantum computation. Diagonal gates of the Clifford hierarchy and ‘nearly diagonal’ semi-Clifford gates are particularly important: they admit efficient gate teleportation protocols that implement these gates with fewer ancillary quantum resources such as magic states. Despite the practical importance of these sets of gates, many questions about their structure remain open; this is especially true in the higher-dimensional qudit setting. Our contribution is to leverage the discrete Stone–von Neumann theorem and the symplectic formalism of qudit stabilizer theory towards extending the results of Zeng et al . (2008) and Beigi & Shor (2010) to higher dimensions in a uniform manner. We further give a simple algorithm for recursively enumerating all gates of the Clifford hierarchy, a simple algorithm for recognizing and diagonalizing semi-Clifford gates, and a concise proof of the classification of the diagonal Clifford hierarchy gates due to Cui et al . (2016) for the single-qudit case. We generalize the efficient gate teleportation protocols of semi-Clifford gates to the qudit setting and prove that every third-level gate of one qudit (of any prime dimension) and of two qutrits can be implemented efficiently. Numerical evidence gathered via the aforementioned algorithms supports the conjecture that higher-level gates can be implemented efficiently.

Cost-optimal single-qubit gate synthesis in the Clifford hierarchy

For universal quantum computation, a major challenge to overcome for practical implementation is the large amount of resources required for fault-tolerant quantum information processing. An important aspect is implementing arbitrary unitary operators built from logical gates within the quantum error correction code. A synthesis algorithm can be used to approximate any unitary gate up to arbitrary precision by assembling sequences of logical gates chosen from a small set of universal gates that are fault-tolerantly performable while encoded in a quantum error-correction code. However, current procedures do not yet support individual assignment of base gate costs and many do not support extended sets of universal base gates. We analysed cost-optimal sequences using an exhaustive search based on Dijkstra’s pathfinding algorithm for the canonical Clifford+Tset of base gates and compared them to when additionally includingZ-rotations from higher orders of the Clifford hierarchy. Two approaches of assigning base gate costs were used. First, costs were reduced toT-counts by recursively applying aZ-rotation catalyst circuit. Second, costs were assigned as the average numbers of raw (i.e. physical level) magic states required to directly distil and implement the gates fault-tolerantly. We found that the average sequence cost decreases by up to54±3%when using theZ-rotation catalyst circuit approach and by up to33±2%when using the magic state distillation approach. In addition, we investigated observed limitations of certain assignments of base gate costs by developing an analytic model to estimate the proportion of sets ofZ-rotation gates from higher orders of the Clifford hierarchy that are found within sequences approximating random target gates.

Four-dimensional toric code with non-Clifford transversal gates

The design of a four-dimensional toric code is explored with the goal of finding a lattice capable of implementing a logical $\mathsf{CCCZ}$ gate transversally. The established lattice is the octaplex tessellation, which is a regular tessellation of four-dimensional Euclidean space whose underlying 4-cell is the octaplex, or hyper-diamond. This differs from the conventional 4D toric code lattice, based on the hypercubic tessellation, which is symmetric with respect to logical $X$ and $Z$ and only allows for the implementation of a transversal Clifford gate. This work further develops the established connection between topological dimension and transversal gates in the Clifford hierarchy, generalizing the known designs for the implementation of transversal $\mathsf{CZ}$ and $\mathsf{CCZ}$ in two and three dimensions, respectively.

Un-Weyl-ing the Clifford Hierarchy

The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being the ubiquitous Pauli group, via the Weyl (i.e., Pauli) expansion of unitaries at these levels. In particular, we characterize the support of the standard Clifford operations on the Pauli group. Since conjugation of a Pauli by a third level unitary produces traceless Hermitian Cliffords, we characterize their Pauli support as well. Semi-Clifford unitaries are known to have ancilla savings in the teleportation model, and we explore their Pauli support via symplectic transvections. Finally, we show that, up to multiplication by a Clifford, every third level unitary commutes with at least one Pauli matrix. This can be used inductively to show that, up to a multiplication by a Clifford, every third level unitary is supported on a maximal commutative subgroup of the Pauli group. Additionally, it can be easily seen that the latter implies the generalized semi-Clifford conjecture, proven by Beigi and Shor (2010). We discuss potential applications in quantum error correction and the design of flag gadgets.

On Optimality of CSS Codes for Transversal T

In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate. Consequently, it is important to understand codes that implement such gates transversally. In this paper, we adopt an algebraic approach to characterize all stabilizer codes for which transversal $T$ and $T^{-1}$ gates preserve the codespace. Our Heisenberg perspective reduces this to a finite geometry problem that translates to the design of certain classical codes. We prove three corollaries: (a) For any non-degenerate $[[ n,k,d ]]$ stabilizer code supporting a physical transversal $T$, there exists an $[[ n,k,d ]]$ CSS code with the same property; (b) Triorthogonal codes are the most general CSS codes that realize logical transversal $T$ via physical transversal $T$; (c) Triorthogonality is necessary for physical transversal $T$ on a CSS code to realize the logical identity. The main tool we use is a recent efficient characterization of certain diagonal gates in the Clifford hierarchy (arXiv:1902.04022). We refer to these gates as Quadratic Form Diagonal (QFD) gates. Our framework generalizes all existing code constructions that realize logical gates via transversal $T$. We provide several examples and briefly discuss connections to decreasing monomial codes, pin codes, generalized triorthogonality and quasitransversality. We partially extend these results towards characterizing all stabilizer codes that support transversal $\pi/2^{\ell}$ $Z$-rotations. In particular, using Ax's theorem on residue weights of polynomials, we provide an alternate characterization of logical gates induced by transversal $\pi/2^{\ell}$ $Z$-rotations on a family of quantum Reed-Muller codes. We also briefly discuss a general approach to analyze QFD gates that might lead to a characterization of all stabilizer codes that support any given physical transversal $1$- or $2$-local diagonal gate.

Unifying the Clifford hierarchy via symmetric matrices over rings

The Clifford hierarchy is a foundational concept for universal quantum computation (UQC). It was introduced to show that UQC can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. (arXiv:1608.06596) recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal gates whose action on a computational basis qudit state is described by a $2^k$-th root of unity raised to a polynomial function of the state, and they established the level of such unitaries in the hierarchy. For qubit systems, we consider $k$-th level diagonal gates that can be described just by quadratic forms of the state over the ring $\mathbb{Z}_{2^k}$ of integers mod $2^k$. These involve symmetric matrices over $\mathbb{Z}_{2^k}$ that can be used to efficiently describe all $2$-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal gates from lower levels. This involves symplectic matrices over $\mathbb{Z}_{2^k}$ and hence our perspective unifies these gates with the binary symplectic framework for Clifford gates. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+$T$ set of gates, perhaps leading to shorter depth circuits. Our results suggest that some non-diagonal gates might be understood by generalizing other binary symplectic matrices to integer rings.

Quantifying magic for multi-qubit operations.

The development of a framework for quantifying 'non-stabilizerness' of quantum operations is motivated by the magic state model of fault-tolerant quantum computation and by the need to estimate classical simulation cost for noisy intermediate-scale quantum (NISQ) devices. The robustness of magic was recently proposed as a well-behaved magic monotone for multi-qubit states and quantifies the simulation overhead of circuits composed of Clifford + T gates, or circuits using other gates from the Clifford hierarchy. Here we present a general theory of the 'non-stabilizerness' of quantum operations rather than states, which are useful for classical simulation of more general circuits. We introduce two magic monotones, called channel robustness and magic capacity, which are well-defined for general n-qubit channels and treat all stabilizer-preserving CPTP maps as free operations. We present two complementary Monte Carlo-type classical simulation algorithms with sample complexity given by these quantities and provide examples of channels where the complexity of our algorithms is exponentially better than previously known simulators. We present additional techniques that ease the difficulty of calculating our monotones for special classes of channels.

Universal MBQC with generalised parity-phase interactions and Pauli measurements

We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form exp⁡(−iπ2nZ⊗Z). When n=2, these are equivalent, up to local Clifford unitaries, to graph states. However, when n>2, their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli Z and X measurements and feed-forward. Even though, for n>2, the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every n>2 this family is capable of producing all Clifford gates and all diagonal gates in the n-th level of the Clifford hierarchy.

Towards Large-scale Functional Verification of Universal Quantum Circuits

We introduce a framework for the formal specification and verification of quantum circuits based on the Feynman path integral. Our formalism, built around exponential sums of polynomial functions, provides a structured and natural way of specifying quantum operations, particularly for quantum implementations of classical functions. Verification of circuits over all levels of the Clifford hierarchy with respect to either a specification or reference circuit is enabled by a novel rewrite system for exponential sums with free variables. Our algorithm is further shown to give a polynomial-time decision procedure for checking the equivalence of Clifford group circuits. We evaluate our methods by performing automated verification of optimized Clifford+T circuits with up to 100 qubits and thousands of T gates, as well as the functional verification of quantum algorithms using hundreds of qubits. Our experiments culminate in the automated verification of the Hidden Shift algorithm for a class of Boolean functions in a fraction of the time it has taken recent algorithms to simulate.

ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity

We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the so-called 'spider' associated with the computational basis, as well as a new arity-N generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike previous graphical calculi, this admits compact encodings of non-linear classical functions. For example, the AND gate can be depicted as a diagram of just 2 generators, compared to ~25 in the ZX-calculus. Consequently, N-controlled gates, hypergraph states, Hadamard+Toffoli circuits, and diagonal circuits at arbitrary levels of the Clifford hierarchy also enjoy encodings with low constant overhead. This suggests that this calculus will be significantly more convenient for reasoning about the interplay between classical non-linear behaviour (e.g. in an oracle) and purely quantum operations. After presenting the calculus, we will prove it is sound and complete for universal quantum computation by demonstrating the reduction of any diagram to an easily describable normal form.

Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates

Stabilizer codes are a simple and successful class of quantum error-correcting codes. Yet this success comes in spite of some harsh limitations on the ability of these codes to fault-tolerantly compute. Here we introduce a new metric for these codes, the disjointness, which, roughly speaking, is the number of mostly non-overlapping representatives of any given non-trivial logical Pauli operator. We use the disjointness to prove that transversal gates on error-detecting stabilizer codes are necessarily in a finite level of the Clifford hierarchy. We also apply our techniques to topological code families to find similar bounds on the level of the hierarchy attainable by constant depth circuits, regardless of their geometric locality. For instance, we can show that symmetric 2D surface codes cannot have non-local constant depth circuits for non-Clifford gates.

Towers of generalized divisible quantum codes

A divisible binary classical code is one in which every code word has weight divisible by a fixed integer. If the divisor is $2^\nu$ for a positive integer $\nu$, then one can construct a Calderbank-Shor-Steane (CSS) code, where $X$-stabilizer space is the divisible classical code, that admits a transversal gate in the $\nu$-th level of Clifford hierarchy. We consider a generalization of the divisibility by allowing a coefficient vector of odd integers with which every code word has zero dot product modulo the divisor. In this generalized sense, we construct a CSS code with divisor $2^{\nu+1}$ and code distance $d$ from any CSS code of code distance $d$ and divisor $2^\nu$ where the transversal $X$ is a nontrivial logical operator. The encoding rate of the new code is approximately $d$ times smaller than that of the old code. In particular, for large $d$ and $\nu \ge 2$, our construction yields a CSS code of parameters $[[O(d^{\nu-1}), \Omega(d),d]]$ admitting a transversal gate at the $\nu$-th level of Clifford hierarchy. For our construction we introduce a conversion from magic state distillation protocols based on Clifford measurements to those based on codes with transversal $T$-gates. Our tower contains, as a subclass, generalized triply even CSS codes that have appeared in so-called gauge fixing or code switching methods.

Distillation with Sublogarithmic Overhead.

It has been conjectured that, for any distillation protocol for magic states for the T gate, the number of noisy input magic states required per output magic state at output error rate ε is Ω[log(1/ε)]. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters ⟦∑[under i=w+1][over m](m/i),∑[under i=0][over w](m/i),∑[under i=w+1][over r+1](r+1/i)⟧ for any integers m>2r, r>w≥0, by puncturing quantum Reed-Muller codes. When m>νr, our code admits a transversal logical gate at the νth level of Clifford hierarchy. In a distillation protocol for magic states at the level ν=3 (T gate), the ratio of input to output magic states is O(log^{γ}(1/ε)), where γ=log(n/k)/log(d)<0.678 for some m, r, w. The smallest code in our family for which γ<1 is on ≈2^{58} qubits.