Where Clifford and Ising quantum computers reach their limits

Can a restricted quantum computer reproduce everything an ordinary computer does efficiently? The question sounds straightforward, but it hides another: how much work is the ordinary computer allowed to do before the quantum calculation begins?

In their November 9, 2026 preprint, Sawyer Dobson and Zhenghan Wang examine that distinction for Clifford quantum computers. They also extend classical simulation results for computers built from Ising anyons, showing that the charge measurements studied remain within an efficiently simulable framework. These are mathematical results about specified computational models, not experimental demonstrations.

Count the classical work too

Clifford circuits use a restricted collection of quantum gates, generated by the Hadamard, quarter-phase and controlled-NOT gates. Starting from suitable states, their evolution can be tracked efficiently on a classical computer using the stabilizer formalism: a compact description based on operators that leave the state unchanged.

But efficient classical simulation does not automatically mean that these circuits can efficiently reproduce every classical computation. Simulating a restricted machine and implementing arbitrary algorithms on it are different tasks.

To examine the reverse direction, the authors vary the abilities of a classical circuit generator—the program that writes the quantum circuit’s instructions. Can it inspect the entire input, or only its length? Can it use polynomial time, or must its working memory grow only logarithmically with input size?

If a polynomial-time generator sees the whole input, the resulting model decides exactly the problems in P, the class solvable in polynomial time by deterministic classical computation. This does not demonstrate that the Clifford circuit performs all that work: the generator can already solve the problem itself.

Restricting that input-aware generator to logarithmic working memory instead yields the class called parity L. Its defining condition concerns whether a nondeterministic logarithmic-space computation has an odd number of accepting branches. Building on earlier work, the paper supplies detailed connections between this class, controlled-NOT circuit simulation and stabilizer circuit simulation.

One circuit for every input of the same size

A different restriction exposes the circuits’ limitations more directly. Here the generator sees only the input length and produces one circuit for all inputs of that size. The actual input enters as a computational-basis state, with additional qubits initialized to zero. The generator cannot inspect circuit outputs to reconstruct the input.

For this model, using a single measured output qubit, the authors characterize the decision problems accessible to polynomial-time-generated Clifford circuits: precisely the polynomial-time problems they call affine languages. Crucially, their definitions require correct circuit decisions and an affine membership rule only for sufficiently large input lengths. Finitely many shorter inputs may be exceptions, including cases where membership is not affine; the circuit family need not decide those exceptions correctly.

For all sufficiently large lengths, the membership rule is an exclusive-or, or XOR, combination of contributions from individual input positions, possibly with a final flip. XOR records whether an odd number of contributions are one. Both the contribution rules and the final flip may depend on input length, but at those lengths the answer cannot contain arbitrary interactions between positions. The definition imposes no such XOR restriction on the finitely many shorter lengths.

Their example is strikingly simple: deciding whether a string begins with two specified letters is outside this model, despite being easy for an ordinary computer. Fix the remaining letters and vary the first two positions between two alternatives. Exactly one of the four strings qualifies. An affine rule must accept an even number of those four combinations at every sufficiently large length. Because this counterexample works at arbitrarily large lengths, allowing short-input exceptions cannot rescue the prefix test.

Encoding matters too. Clifford circuits cannot freely perform every classical conversion between representations. The authors show that a one-hot encoding—giving each alphabet symbol its own bit position—provides a common way to analyze which languages qualify.

For logarithmic-space-generated circuit families, they establish upper and lower bounds rather than an exact characterization. The remaining gap is connected to unresolved relationships between logarithmic-space computation and parity L.

Measuring anyons does not escape the framework

The paper’s other main result concerns Ising anyons, mathematical models of quasiparticles relevant to topological quantum computing. Information is encoded in their collective states; exchanging them along braided paths performs quantum operations.

Ising braiding was already known to admit efficient classical simulation. The authors extend that conclusion to charge measurements, including interferometric measurements of collections of anyons.

Using the Ising model’s fusion and braiding rules, they show that every such interferometric measurement becomes a projective Pauli measurement in the dense qubit encoding. Measurements of separated collections are handled by bringing them together through braiding and undoing that rearrangement afterward.

Consequently, the modeled operations remain within the Gottesman–Knill simulation framework: Clifford operations and Pauli measurements starting from stabilizer states can be simulated efficiently by a probabilistic classical computer.

What remains unresolved

The authors do not settle whether appropriately defined Ising and Clifford computers have equal computational power. Classical simulability alone does not establish that equivalence.

Nor do they separate P from PSPACE, the class of problems solvable with polynomial working memory. They explain a conditional route: if every problem in P reduced to stabilizer simulation using logarithmic-space processing, a known small-space bound would imply P differs from PSPACE. Proving that premise remains open.

The central lesson is therefore model-specific: measuring the studied Ising system does not defeat classical simulation, and attributing computational power to Clifford hardware requires carefully separating its work from the classical controller’s.