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Wolfram`QuantumFramework`

PauliStabilizer

PauliStabilizer []
	represents an empty stabilizer state, equivalent to the single-qubit register \|0〉 .

PauliStabilizer [n]
	represents the n-qubit \|0·s0〉 register state, with stabilizers Z i .

PauliStabilizer [{ stab 1 , stab 2 ,…}]
	constructs a stabilizer state from a list of Pauli strings.

PauliStabilizer [{stabs},{destabs}]
	constructs a stabilizer state with explicit stabilizers and destabilizers.

PauliStabilizer [name]
	returns a named stabilizer state. Names: "5QubitCode" , "SteaneCode" , "9QubitCode" , their "·1" siblings, and "Random" .

PauliStabilizer ["Random",n]
	returns a uniformly random n-qubit Clifford state via the Mallows sampler.

PauliStabilizer [qs]
	converts a QuantumState , QuantumOperator , or QuantumCircuitOperator to a stabilizer state (Clifford only).

Details and Options

▪

PauliStabilizer

encodes an n-qubit pure stabilizer state by a tableau of Pauli generators plus signs, requiring

)

memory.

▪

The internal representation is an Association

"Tableau"t,"Signs"s

where t is a rank-3 binary array of shape

{2,n,2n}

(X-block, Z-block; n qubits;

rows: destabilizers then stabilizers) and s is a length-

list of

±1

signs.

▪

Equivalent serialization keys are also accepted:

"Phase"

1-Signs/2

) and

"Matrix"

(the flat

2n×2n

binary tableau).

▪

Stabilizer rows are the n parity-check generators of the codespace. Destabilizer rows complete the symplectic basis and are required for measurement and

"Dagger"

. String-list constructors auto-generate destabilizers via

Reverse

; supply them explicitly with the two-list form to avoid singular fixtures.

▪

Named codes follow the convention that

"5QubitCode"

encodes

〉

(the +1 eigenstate of logical

), and the

"·1"

sibling encodes

〉

▪

PauliStabilizer

["Random",n]

uses the Bravyi-Maslov / Koenig-Smolin (arXiv:1406.2170) Mallows sampler for uniform sampling over the

-qubit Clifford group modulo Paulis.

▪

A stabilizer state ps supports two access patterns: properties via

ps["Property"]

(e.g.

"Stabilizers"

"Tableau"

"State"

) and methods via

ps["Method",…]

(e.g. Clifford gates, measurement,

"InnerProduct"

). The full list of accepted strings is available via

ps["Properties"]

▪

As an operator, a constructed ps is applied by string dispatch:

ps[method,…]

applies a Clifford gate, measurement, or property accessor, and

ps[

]

composes two stabilizer states (apply

first, then ps).

▪

Clifford gate updates run in

)

time using the Aaronson-Gottesman tableau (arXiv:quant-ph/0406196). Single-qubit gates:

"H"

"S"

†

"X"

"Y"

"Z"

"V"

. Two-qubit gates:

"CNOT"

(or

"CX"

"CZ"

"SWAP"

▪

Non-Clifford gates

"P"[θ]

"T"

, and

†

return a

StabilizerFrame

(a coherent superposition of stabilizer states) rather than a

PauliStabilizer

. The frame closes under further Clifford gates.

▪

Symbolic measurement (Fang-Ying 2023, arXiv:2311.03906) via

"SymbolicMeasure"

allocates a fresh

s[k]

outcome symbol per non-deterministic measurement, yielding a single PauliStabilizer with symbolic phase. Resolve via

"SubstituteOutcomes"

"SampleOutcomes"

▪

The option

Method

ps["InnerProduct",other]

selects between exact state-vector materialization (

"Direct"

, default) and the closed-form Garcia-Markov-Cross algorithm (

"ClosedForm"

, arXiv:1210.6646).

	Option	Default value	Description
	"Method"	"Direct"	the algorithm to use when computing the inner product

▪

Possible

"Method"

values:

Value	Default	Description
"Direct"	default	materialize state vectors and compute exactly ( O( n 2 ) , recovers complex phase)
"ClosedForm"		closed-form O( 3 n ) magnitude only, no complex phase (concrete signs only)