A complete theory of the Clifford commutant
read the original abstract
The Clifford group plays a central role in quantum information science. It is the building block for many error-correcting schemes and matches the first three moments of the Haar measure over the unitary group-a property that is essential for a broad range of quantum algorithms, with applications in pseudorandomness, learning theory, benchmarking, and entanglement distillation. At the heart of understanding many properties of the Clifford group lies the Clifford commutant: the set of operators that commute with $k$-fold tensor powers of Clifford unitaries. Previous understanding of this commutant has been limited to relatively small values of $k$, constrained by the number of qubits $n$. In this work, we develop a complete theory of the Clifford commutant. Our first result provides an explicit orthogonal basis for the commutant and computes its dimension for arbitrary $n$ and $k$. We also introduce an alternative and easy-to-manipulate basis formed by isotropic sums of Pauli operators. We show that this basis is generated by products of permutations, which generate the unitary group commutant, and at most three other operators. Additionally, we develop a graphical calculus allowing a diagrammatic manipulation of elements of this basis. These results enable a wealth of applications: among others, we characterize all measurable magic measures and identify optimal strategies for stabilizer property testing, whose success probability also offers an operational interpretation to stabilizer entropies. Finally, we show that these results also generalize to multi-qudit systems with prime local dimension.
This paper has not been read by Pith yet.
Forward citations
Cited by 10 Pith papers
-
Coherent-State Propagation: A Computational Framework for Simulating Bosonic Quantum Systems
Coherent-state propagation enables quasi-polynomial classical simulation of bosonic circuits with logarithmically many Kerr gates at exponentially small trace-distance error, with polynomial runtime in the weak-nonlin...
-
Cloning is as Hard as Learning for Stabilizer States
For n-qubit stabilizer states the optimal sample complexity of approximate cloning is Θ(n), matching the complexity of learning.
-
Demonstrating an unconditional separation between quantum and classical information resources
Demonstrates a task solvable with 12 qubits but requiring 62-382 classical bits of memory, yielding unconditional quantum information supremacy on a trapped-ion processor.
-
Operational interpretation of the Stabilizer Entropy
The stabilizer Rényi entropy governs the exponential rate at which Clifford orbits become indistinguishable from Haar-random states and sets the optimal distinguishability from stabilizer states in property testing.
-
Geometry of Free Fermion Commutants
The k-commutant of free fermions is the Grassmannian manifold of fermionic Gaussian states on 2k sites, exposing a real-replica space duality.
-
Nonstabilizerness and Error Resilience in Noisy Quantum Circuits
Amplitude damping generates nonstabilizerness in qubit systems unlike depolarizing noise, with local injection washed out collectively after encoding, decoding, and postselection.
-
Non-Clifford Cost of Random Unitaries
Rigorous bounds establish that t = Theta(k^2) non-Clifford gates are necessary and sufficient for frame-potential approximation to unitary k-designs while t = Theta(nk) suffices for relative-error k-designs.
-
Stabilizer entropy is trustworthy for mixed states
A linear Stabilizer Entropy acts as a non-stabilizerness monotone with overwhelming probability for mixed states under non-adaptive Clifford channels on flat stabilizer states, with violation probabilities decaying ex...
-
Taming Trotter Errors with Quantum Resources
Higher entanglement entropy reduces variance of Trotter errors and higher magic reduces kurtosis, making error distributions more robust in quantum simulation.
-
Lecture Notes on Replica Tensor Networks for Random Quantum Circuits
Lecture notes and accompanying library teach replica tensor network methods to compute circuit-averaged observables in random quantum circuits by mapping them to classical statistical mechanics models.
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.