Orthogonal Clifford Porter-Thomas Distribution

• The orthogonal Clifford Porter-Thomas distribution is defined for output probability overlaps in rebit quantum circuits, exhibiting discrete support and super-Gaussian tail decay.
• It employs real Clifford Weingarten calculus to derive moments that distinguish its statistical properties from Haar-random and complex Clifford distributions.
• The distribution is pivotal in error-corrected architectures and benchmarks randomized circuits, guiding tensor-network simulations with minimal magic state requirements.

The orthogonal Clifford Porter-Thomas (OCPT) distribution characterizes the statistical behavior of output probability overlaps from random real (orthogonal) Clifford circuits acting on pure quantum states. Unlike the classical Porter-Thomas law arising from Haar-random unitaries, the OCPT law defines a genuinely new universality class, with discrete support, super-Gaussian tail decay, and moments determined via the real Clifford Weingarten calculus. Its physical relevance spans error-corrected architectures over rebits, randomized benchmarking, and tensor-network constructions using orthogonal Clifford tensors. The OCPT law emerges universally in local real Clifford circuits at logarithmic depth, with its statistical properties sharply distinct from both the unitary-Haar and complex Clifford cases (Magni et al., 17 Dec 2025).

1. Definition and Explicit Formulation

For an $N$-qubit real stabilizer state $|\psi\rangle \in \R\mathrm{Stab}_N$, amplitudes are supported on computational basis vectors with squared modulus $2^{-g}$, where $0\leq g \leq N$ is the support rank. The scaled overlap $w$ with the reference vector $|0^N\rangle$ takes values $$w=2^N |\langle 0^N|\psi\rangle|^2=2^{N-g}\equiv 2^n$$ with $n=N-g$.

Enumerating all real stabilizer states leads to the exact probability mass function: $$\P_\R(n) = \frac{1}{(-1;2^{-1})_\infty} \frac{2^{-n(n+1)/2} \ 2^n}{(2^{-1};2^{-1})_n} \qquad n=0,1,2,\dots$$ where $(a;q)_n = \prod_{j=0}^{n-1}(1-a q^j)$ is the $q$-Pochhammer symbol and $(-1;2^{-1})_\infty = \prod_{j\ge0}(1+2^{-j-1})$. The distribution decays super-Gaussianly with increasing $n$, in sharp contrast to the exponential decay found in unitary-Haar scenarios (Magni et al., 17 Dec 2025).

2. Derivation via Real Clifford Weingarten Calculus

All moments and the distribution itself are captured by calculating $k$-th moments $E[|\langle y|\psi\rangle|^{2k}]$ over the real Clifford group $\R_N$. The formalism employs the Schur–Weyl duality for the real Clifford group: $\E_{C\in\R_n}[(C\otimes C^*)^{\otimes k}] = \sum_{\sigma,\pi \in \Xi_k} \Wg_{\sigma,\pi}(2^n) |\sigma\rangle\!\rangle \langle\!\langle \pi|$ Here, $\Xi_k$ indexes the commutant algebra for k replicas, and $\Wg_{\sigma,\pi}$ is the real-Clifford Weingarten matrix, a pseudo-inverse of the corresponding Gram matrix. The purity of stabilizer states under this formalism is $0 \leq \zeta_\sigma(|\psi\rangle) \leq 1$ ($\zeta_\sigma=1$ for real stabilizers), and summation identities for Weingarten functions determine the precise normalization and support structure (Magni et al., 17 Dec 2025).

Inverse Participation Ratios (IPRs) and overlap moments become: $I_k^\R = 2^{(1-k)N}(-2;2)_{k-1}(-2^{-N+1};2)_{k-1}$ with $(-2;2)_m = \prod_{j=0}^{m-1}(1+2^j)$.

3. Statistical Characteristics

The OCPT distribution exhibits unique statistical properties:

Quantity	OCPT Value	Remarks
Mean	$I_1^\R = 1$	$\langle w \rangle = 1$
Variance	$I_2^\R = 3 \ 2^{-N}(1 + O(2^{-N}))$	$\mathrm{Var}(w) = 2$
Higher Moments	$E[w^k] = d^k I_k^\R \to (-2;2)_{k-1}$ (large $N$)	Grows as $\prod_{m=1}^{k-1}(1 + 2^m)$
Tail Decay	$\sim 2^{-n(n+1)/2}$ for large $n$	Super-Gaussian, dominant tail

Moments up to third order agree with those for orthogonal-Haar ensembles, reflecting that real Cliffords constitute a 3-design. For $k>3$, deviations appear, distinguishing OCPT from Haar statistics (Magni et al., 17 Dec 2025).

4. Comparison to Other Universality Classes

The OCPT law sits in a hierarchy of statistical distributions for quantum circuit outputs:

• Unitary-Haar: $\P_\U(w) \to e^{-w}$; continuous, exponential tail, moments $I_k^{\C\mathrm{Haar}} = k!/\prod_{m=0}^{k-1}(d + m)$.
• Orthogonal-Haar: $\P_\O(w) \propto w^{-1/2}e^{-w/2}$; chi-squared, continuous, moments $$I_k^{\R\mathrm{Haar}} = (2k-1)!!/\prod_{m=0}^{k-1}(d + 2m)$$.
• Complex-Clifford: Discrete “Clifford-PT,” supported on support rank $g$, moments $I_k^{\C\mathrm{Stab}} = d^{1-k}(-1;2)_{k-1}(-d^{-1};2)_{k-1}$.
• Orthogonal-Clifford (OCPT): Discrete, super-Gaussian decay, moments as above.

The OCPT law fundamentally differs from Porter-Thomas, exhibiting discrete support on $n$ and Pochhammer-derived normalization. It arises from the restriction to real quantum gates and so quantifies “rebit-chaotic” behavior distinct from complex stabilizer or Haar ensembles (Magni et al., 27 Feb 2025, Magni et al., 17 Dec 2025).

5. Circuit Depths and Resource Hierarchies

Local architectures of random orthogonal Clifford gates (e.g., brickwork or glued circuits) anticoncentrate to the OCPT law in depth $t \sim O(\log N)$. Random matrix product states (RMPS) with bond dimension $\chi \sim \mathrm{poly}(N)$ also recover this regime. The process is robust and does not require non-Clifford resources (Magni et al., 17 Dec 2025).

Resource hierarchies for driving the output distribution towards Haar statistics:

• Real-magic doping: Injecting $r = O(\log N)$ copies of $|H\rangle$ recovers real Haar moments.
• Complex-magic doping: Polylogarithmic $|T\rangle$ states drives towards full unitary Haar Porter-Thomas.
• Imaginary resource injection: A single imaginary stabilizer $|\pm i\rangle$ suffices for complex Clifford design (unitary Clifford statistics).

A plausible implication is that the minimal resource cost for accessing unitary Clifford or Haar-like behavior is sharply stratified—full Haar randomness is reached only with polylogarithmic “magic” state injection, while complex Clifford behavior occurs with a single imaginary injection (Magni et al., 17 Dec 2025, Magni et al., 27 Feb 2025).

6. Physical Interpretation and Applications

The OCPT distribution provides a baseline for “minimal” universality when circuits use real gates, with distinctive implications for benchmarking, error correction, and tensor network simulation. It quantifies the spreading of amplitudes in rebit-Clifford circuits, which, due to their structure, populate supports on sizes $2^g$ and yield discrete statistics in $n=N-g$. Its emergence at logarithmic circuit depth signals rapid approach to real-chaotic statistics without non-Clifford resource injection.

The mapped resource hierarchy elucidates the distinct functional roles of real-magic, complex-magic, and imaginarity in quantum state design. In architectures prioritizing rebits or error correction, OCPT sets the standard for “realizable” randomness and benchmarks pseudo-magic schemes (Magni et al., 27 Feb 2025, Magni et al., 17 Dec 2025).

7. Summary and Outlook

The orthogonal Clifford Porter-Thomas distribution,

$$\P_\R(n) = \frac{1}{(-1;2^{-1})_\infty} \frac{2^{-n(n+1)/2} \ 2^n}{(2^{-1};2^{-1})_n}$$

is the universal statistical law for output overlaps in real-Clifford circuits at logarithmic depth. Its moments,

$$I_k^\R = 2^{(1-k)N}(-2;2)_{k-1}(-2^{-N+1};2)_{k-1} \rightarrow d^{1-k}(-2;2)_{k-1}$$

in the thermodynamic limit, are super-Gaussian in $n$, discrete, and break with standard exponential Porter-Thomas form. The OCPT law thereby defines a new universality class for quantum circuit statistics, forming a critical reference point in randomized benchmarking, tensor-network state design, and resource quantification for quantum computational advantage (Magni et al., 27 Feb 2025, Magni et al., 17 Dec 2025).