Clifford Group Research Articles

On the stabilizer formalism and its generalization

Abstract The standard stabilizer formalism provides a setting to show that quantum computations restricted to operations within the Clifford group are classically efficiently simulable: this is the content of the well-known Gottesman–Knill theorem. This work analyzes the mathematical structure behind this theorem to find possible generalizations and derivation of constraints required for constructing a non-trivial generalized Clifford group. We prove that if the closure of the stabilizing set is dense in the set of SU(d) transformations, then the associated Clifford group is trivial, consisting only of local gates and permutations of subsystems. This result demonstrates the close relationship between the density of the stabilizing set and the simplicity of the corresponding Clifford group. We apply the analysis to investigate stabilization with binary observables for qubits and find that the formalism is equivalent to the standard stabilization for a low number of qubits. Based on the observed patterns, we conjecture that a large class of generalized stabilizer states are equivalent to the standard ones. Our results provide better insights into the structure of Gottesman–Knill-type results, consequently allowing us to draw a sharper line between quantum and classical computation.

Revealing symmetries in quantum computing for many-body systems

Abstract We develop a method to deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan-Wigner form in which they can act on multi-qubit states. 
Symmetries, such as point-group symmetries in molecules, are apparent in the standard second quantized form of the Hamiltonian.
They are, however,
masked when the Hamiltonian is translated into a Pauli matrix representation required for its operation on qubits.
To reveal these symmetries we prove a general theorem that provides a straightforward method to calculate the transformation of Pauli tensor strings under symmetry operations.
They are a subgroup of the Clifford group transformations and induce a corresponding group representation inside the symplectic matrices.
We finally give a simplified derivation of an affine qubit encoding scheme which allows for the removal of qubits due to Boolean symmetries and thus reduces effort in quantum computations
for many-body systems.

Clifford orbits and stabilizer states

Abstract Stabilizer states serve as “classical objects” in the stabilizer formalism of quantum theory, and play an important role in quantum error correction, fault-tolerant quantum computation, and quantum communication. They provide an efficient description of many basic features of quantum theory and exhibit a rich structure. For prime dimensional systems, they may be defined by two quite different yet equivalent ways: The first is via stabilizer groups (maximal Abelian subgroups of the discrete Heisenberg-Weyl group). The second is via the orbits of the Clifford group acting on any computational basis state. However, in a general dimensional system, this equivalence breaks down, and consequently, it is desirable to classify the difference and relation between the above two approaches to stabilizer states. In this work, we show that these two approaches are equivalent if and only if the system dimension is square-free (i.e., has no square factor). Furthermore, we completely clarify the relation between the Clifford orbits and stabilizer states in an arbitrary dimensional system, derive the explicit expressions of the Clifford orbits and determine their cardinalities.

Quantum computation from dynamic automorphism codes

We propose a new model of quantum computation comprised of low-weight measurement sequences that simultaneously encode logical information, enable error correction, and apply logical gates. These measurement sequences constitute a new class of quantum error-correcting codes generalizing Floquet codes, which we call dynamic automorphism (DA) codes. We construct an explicit example, the DA color code, which is assembled from short measurement sequences that can realize all 72 automorphisms of the 2D color code. On a stack of N triangular patches, the DA color code encodes N logical qubits and can implement the full logical Clifford group by a sequence of two- and, more rarely, three-qubit Pauli measurements. We also make the first step towards universal quantum computation with DA codes by introducing a 3D DA color code and showing that a non-Clifford logical gate can be realized by adaptive two-qubit measurements.

Group theoretical classification of SIC-POVMs

The Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) are known to exist in all dimensions ⩽ 151 and many larger dimensions as high as 39604 . All known solutions with the exception of the Hoggar solutions are covariant with respect to the Weyl-Heisenberg group and in the case of dimension 3, it has been proven that all SIC-POVMs are Weyl-Heisenberg group covariant. In this work, we explore this by SIC-POVMs in dimensions 4–7 without the assumption of group covariance. We introduce two functions with which SIC-POVM Gram matrices can be generated without the group covariance constraint, and analytically show that the SIC-POVM Gram matrices exist on critical points of the surfaces defined by the two functions on a subspace of Hermitian matrices. In dimensions 4–7, all known SIC-POVM Gram matrices lie in disjoint continuous sets of solution. Thus, we define an equivalent class that relates Gram matrices based on the trivial symmetries of the two functions. In dimensions 4–7, we generated {1.7×106,1.1×105,169,50} Gram matrices, respectively. For each of the Gram matrices, we generate the symmetry group of their respective disjoint sets. In all cases, the symmetry group was isomorphic to a subgroup of the Clifford group containing the Weyl-Heisenberg group matrices and the order-3 unitaries. In dimensions 4–6, All constructed solutions belonged to a single equivalence class of Gram matrices whereas in dimension 7, we find two distinct families of SIC-POVMs. In dimensions 4 and 5, the absence of new classes of SIC-POVM strongly suggests that the functions do not have a non-trivial symmetry. Furthermore, in all the dimensions tested, the only equivalence classes found correspond to the distinct stabilizers of fiducial vectors of each dimension.

Low-depth Clifford circuits approximately solve MaxCut

We introduce a quantum-inspired approximation algorithm for MaxCut based on low-depth Clifford circuits. We start by showing that the solution unitaries found by the adaptive quantum approximation optimization algorithm (ADAPT-QAOA) for the MaxCut problem on weighted fully connected graphs are (almost) Clifford circuits. Motivated by this observation, we devise an approximation algorithm for MaxCut, ADAPT-Clifford, that searches through the Clifford manifold by combining a minimal set of generating elements of the Clifford group. Our algorithm finds an approximate solution of MaxCut on an N-vertex graph by building a depth O(N) Clifford circuit. The algorithm has runtime complexity O(N2) and O(N3) for sparse and dense graphs, respectively, and space complexity O(N2), with improved solution quality achieved at the expense of more demanding runtimes. We implement ADAPT-Clifford and characterize its performance on graphs with positive and signed weights. The case of signed weights is illustrated with the paradigmatic Sherrington-Kirkpatrick model, for which our algorithm finds solutions with ground-state mean energy density corresponding to ∼94% of the Parisi value in the thermodynamic limit. The case of positive weights is investigated by comparing the cut found by ADAPT-Clifford with the cut found with the Goemans-Williamson (GW) algorithm. For both sparse and dense instances we provide copious evidence that, up to hundreds of nodes, ADAPT-Clifford finds cuts of lower energy than GW. Published by the American Physical Society 2024

Fold-Transversal Clifford Gates for Quantum Codes

We generalize the concept of folding from surface codes to CSS codes by considering certain dualities within them. In particular, this gives a general method to implement logical operations in suitable LDPC quantum codes using transversal gates and qubit permutations only.To demonstrate our approach, we specifically consider a [[30, 8, 3]] hyperbolic quantum code called Bring's code. Further, we show that by restricting the logical subspace of Bring's code to four qubits, we can obtain the full Clifford group on that subspace.

Quantum circuit optimization of an integer divider

Efficient arithmetic operations in quantum circuits play a vital role in the implementation of quantum algorithms. Quantum circuits constructed exclusively using gates of the Clifford+T group are compatible with error detection and correction codes available in the quantum literature. However, the T gate, a member of this group, has a higher cost compared to other gates, making it crucial to minimize its usage to reduce circuit expenses. While the T gate cannot be entirely avoided since the Clifford group is not a universal set of gates, circuit optimization can effectively reduce the number of T gates required for implementation. In this work, we present a novel divider circuit for quantum computing that focuses on reducing the number of T gates while maintaining a reasonable number of qubits for this type of operation. To achieve this, we introduce variants of minor circuits, including a comparator and two types of subtractors. These circuits are based on published literature but undergo modifications to optimize their resource utilization for performing the division operation. The obtained results demonstrate that the proposed divider circuit outperforms other currently published divider circuits in terms of T gate usage, highlighting its efficiency and potential practicality in quantum algorithms.

On special Clifford groups and their characteristic classes

On special Clifford groups and their characteristic classes

Clifford orbits from cayley graph quotients

We describe the structure of the $n$-qubit Clifford group $\mathcal{C}_n$ via Cayley graphs, whose vertices represent group elements and edges represent generators. In order to obtain the action of Clifford gates on a given quantum state, we introduce a quotient procedure. Quotienting the Cayley graph by the stabilizer subgroup of a state gives a reduced graph which depicts the state's Clifford orbit. Using this protocol for $\mathcal{C}_2$, we reproduce and generalize the reachability graphs introduced in \cite{Keeler2022}. Since the procedure is state-independent, we extend our study to non-stabilizer states, including the W and Dicke states. Our new construction provides a more precise understanding of state evolution under Clifford circuit action.

Clifford Group and Unitary Designs under Symmetry

Clifford Group and Unitary Designs under Symmetry

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Clifford group is not a semidirect product in dimensions N divisible by four

The paper is devoted to projective Clifford groups of quantum N-dimensional systems (with configuration space ). Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann–Knill theorem). However, it may serve as a cornerstone of full quantum computation. As to its group structure it is well-known that—in N-dimensional quantum mechanics—the Clifford group is a natural semidirect product provided the dimension N is an odd number. For even N special results on the Clifford groups are scattered in the mathematical literature, but they mostly do not concern the semidirect structure. Using appropriate group presentation of it is proved that for even N the projective Clifford groups are not natural semidirect products if and only if N is divisible by four.

Scalable fast benchmarking for individual quantum gates with local twirling

With the development of controllable quantum systems, fast and practical characterization of multi-qubit gates has become essential for building high-fidelity quantum computing devices. The usual way to fulfill this requirement via randomized benchmarking demands complicated implementation of numerous multi-qubit twirling gates. How to efficiently and reliably estimate the fidelity of a quantum process remains an open problem. This work thus proposes a character-cycle benchmarking protocol and a character-average benchmarking protocol using only local twirling gates to estimate the process fidelity of an individual multi-qubit operation. Our protocols were able to characterize a large class of quantum gates including and beyond the Clifford group via the local gauge transformation, which forms a universal gate set for quantum computing. We demonstrated numerically our protocols for a non-Clifford gate—controlled- ( T X ) and a Clifford gate—five-qubit quantum error-correcting encoding circuit. The numerical results show that our protocols can efficiently and reliably characterize the gate process fidelities. Compared with the cross-entropy benchmarking, the simulation results show that the character-average benchmarking achieves three orders of magnitude improvements in terms of sampling complexity.

Entropic lens on stabilizer states

The $n$-qubit stabilizer states are those left invariant by a ${2}^{n}$-element subset of the Pauli group. The Clifford group is the group of unitaries which take stabilizer states to stabilizer states; a physically motivated generating set, the Hadamard, phase, and controlled-not (cnot) gates which comprise the Clifford gates, impose a graph structure on the set of stabilizers. We explicitly construct these structures, the ``reachability graphs,'' at $n\ensuremath{\le}5$. When we consider only a subset of the Clifford gates, the reachability graphs separate into multiple, often complicated, connected components. Seeking an understanding of the entropic structure of the stabilizer states, which is ultimately built up by cnot gate applications on two qubits, we are motivated to consider the restricted subgraphs built from the Hadamard and cnot gates acting on only two of the $n$ qubits. We show how the two subgraphs already present at two qubits are embedded into more complicated subgraphs at three and four qubits. We argue that no additional types of subgraph appear beyond four qubits, but that the entropic structures within the subgraphs can grow progressively more complicated as the qubit number increases. Starting at four qubits, some of the stabilizer states have entropy vectors which are not allowed by holographic entropy inequalities. We comment on the nature of the transition between holographic and nonholographic states within the stabilizer reachability graphs.

Efficient simulation of Gottesman-Kitaev-Preskill states with Gaussian circuits

We study the classical simulatability of Gottesman-Kitaev-Preskill (GKP) states in combination with arbitrary displacements, a large set of symplectic operations and homodyne measurements. For these types of circuits, neither continuous-variable theorems based on the non-negativity of quasi-probability distributions nor discrete-variable theorems such as the Gottesman-Knill theorem can be employed to assess the simulatability. We first develop a method to evaluate the probability density function corresponding to measuring a single GKP state in the position basis following arbitrary squeezing and a large set of rotations. This method involves evaluating a transformed Jacobi theta function using techniques from analytic number theory. We then use this result to identify two large classes of multimode circuits which are classically efficiently simulatable and are not contained by the GKP encoded Clifford group. Our results extend the set of circuits previously known to be classically efficiently simulatable.

Some criteria for stably birational equivalence of quadratic forms

Let φ and ψ be quadratic forms over a field K of characteristic different from 2. In this paper, we give a criterion for isotropy of φ over the function field of ψ in terms of representations and we apply it to stably birational equivalence of φ and ψ. Then we use this criterion to investigate the case of stably birational equivalence of multiples of Pfister forms. Finally, we give a criterion of stably birational equivalence of quadratic forms in terms of isomorphisms of quotients of special Clifford groups by the kernel of the spinor norm.

Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates

Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full n-qubit group, one often resorts to t-designs. Unitary t-designs mimic the Haar-measure up to t-th moments. It is known that Clifford operations can implement at most 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject O(t^{4}log ^{2}(t)log (1/varepsilon )) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an varepsilon -approximate t-design. Strikingly, the number of non-Clifford gates required is independent of the system size – asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the t-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.

Magic-state resource theory for the ground state of the transverse-field Ising model

Ground states of quantum many-body systems are both entangled and possess a kind of quantum complexity as their preparation requires universal resources that go beyond the Clifford group and stabilizer states. These resources - sometimes described as magic - are also the crucial ingredient for quantum advantage. We study the behavior of the stabilizer R\'enyi entropy in the integrable transverse field Ising spin chain. We show that the locality of interactions results in a localized stabilizer R\'enyi entropy in the gapped phase thus making this quantity computable in terms of local quantities in the gapped phase, while measurements involving $L$ spins are necessary at the critical point to obtain an error scaling with $O(L^{-1})$.

Integrable spin chains and the Clifford group

We construct new families of spin chain Hamiltonians that are local, integrable, and translationally invariant. To do so, we make use of the Clifford group that arises in quantum information theory. We consider translation invariant Clifford group transformations that can be described by matrix product operators (MPOs). We classify translation invariant Clifford group transformations that consist of a shift operator and an MPO of bond dimension two—this includes transformations that preserve locality of all Hamiltonians and those that lead to non-local images of particular operators but, nevertheless, preserve locality of certain Hamiltonians. We characterize translation invariant Clifford group transformations that take single-site Pauli operators to local operators on at most five sites—examples of Quantum Cellular Automata—leading to a discrete family of Hamiltonians that are equivalent to the canonical XXZ model under such transformations. For spin chains solvable by the algebraic Bethe ansatz, we explain how conjugating by an MPO affects the underlying integrable structure. This allows us to relate our results to the usual classifications of integrable Hamiltonians. We also treat the case of spin chains solvable by free fermions.