Even-depth permuted brickwork fails positive semidefiniteness

Abstract

For four sites, every local dimension q at least two and every even depth at least two give a permuted-brickwork Haar moment with a strictly negative quadratic form.

Definition 1.1 (The three complete matchings).

$$\forall m\in \mathrm{Fin}\left(3\right),\mathrm{matching}\left(m\right)=\mathrm{ite}\left(m=0,![(0,1),(2,3)],\mathrm{ite}\left(m=1,![(0,2),(1,3)],![(0,3),(1,2)]\right)\right)$$

Formalization. D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.matching (✓ std3).

Citation. Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Commentary.

Section 5.2 (page 12): “Suppose we draw a random two-sided matching of the sites, then apply a layer of Haar-random 2-site gates to those pairs in parallel.” The source sites are numbered 0,1,2,3 here. Matching indices 0,1,2 denote A,B,C, respectively.

Definition 1.2 (Restricting the replicas to one edge).

$$\forall q\in \mathbb{N},\forall p\in (\mathrm{Fin}\left(4\right)\times \mathrm{Fin}\left(4\right)),\forall r\in (((\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right))\times (\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right)))\times ((\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right))\times (\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right)))),\mathrm{restrictReplica}\left(p,r\right)=(((r.1.1\left(p.1\right),r.1.1\left(p.2\right)),(r.1.2\left(p.1\right),r.1.2\left(p.2\right))),((r.2.1\left(p.1\right),r.2.1\left(p.2\right)),(r.2.2\left(p.1\right),r.2.2\left(p.2\right))))$$

Formalization. D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.restrictReplica (✓ std3).

Source. Repository-derived.

Acknowledgement. Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Commentary.

Each of the four replicas is restricted to the two endpoints in their displayed order. Their order remains barred one, barred two, unbarred one, unbarred two.

Definition 1.3 (A layer of independent Haar gates).

$$\forall q\in \mathbb{N},\forall m\in \mathrm{Fin}\left(3\right),\forall r\in (((\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right))\times (\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right)))\times ((\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right))\times (\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right)))),\forall c\in (((\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right))\times (\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right)))\times ((\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right))\times (\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right)))),\mathrm{layerMoment}\left(q,m\right)\left(r,c\right)=\prod _{e:\mathrm{Fin}\left(2\right)}(\mathrm{haarAverage}\left((\mathrm{Fin}\left(q\right)\times \mathrm{Fin}\left(q\right))\right)\left(\mathrm{restrictReplica}\left(\mathrm{matching}\left(m\right)\left(e\right),r\right),\mathrm{restrictReplica}\left(\mathrm{matching}\left(m\right)\left(e\right),c\right)\right))$$

Formalization. D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.layerMoment (✓ std3).

Citation. Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Commentary.

The two disjoint pairs in a matching receive independent normalized Haar gates on Matrix.unitaryGroup (Fin q times Fin q) over the complex numbers. The matrix entry is the product of their literal four-factor expectations.

Definition 1.4 (A circuit word).

$$\forall q\in \mathbb{N},\forall d\in \mathbb{N},\forall w\in (\mathrm{Fin}\left(d\right)\to \mathrm{Fin}\left(3\right)),\mathrm{wordMoment}\left(q,d,w\right)=List.prod\left(List.map\left(\lambda i:\mathrm{Fin}\left(d\right),\mathrm{layerMoment}\left(q,w\left(i\right)\right),List.reverse\left(List.finRange\left(d\right)\right)\right)\right)$$

Formalization. D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.wordMoment (✓ std3).

Citation. Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Commentary.

The circuit product is U_(d-1) … U_0. The layer-moment matrices therefore multiply in reverse index order, as in the actual circuit.

Definition 1.5 (The connected-block condition on four sites).

$$\forall d\in \mathbb{N},\forall w\in (\mathrm{Fin}\left(d\right)\to \mathrm{Fin}\left(3\right)),\mathrm{NoRepeat}\left(d,w\right)=\left(\forall i\in \mathbb{N},(i+1<d)\Rightarrow (w\left(Fin.mk\left(i,\cdot \right)\right)\ne w\left(Fin.mk\left(i+1,\cdot \right)\right))\right)$$

Formalization. D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.NoRepeat (✓ std3).

Citation. Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Commentary.

Section 5.2 (page 12): “Suppose we draw layers as in the parallel complete-graph architecture, except that we require each adjacent pair of layers to form a connected block.” On four sites, two equal matchings have two connected components, while any two distinct matchings have a four-cycle as their union. Thus connected union is exactly the condition that adjacent matching indices differ. The Fin.mk arguments retain the bounds supplied by i+1 < d.

Definition 1.6 (The admissible words).

$$\forall d\in \mathbb{N},\mathrm{admissibleWords}\left(d\right)=Finset.filter\left(\mathrm{NoRepeat}\left(d\right),(Finset.univ:\mathrm{Finset}\left((\mathrm{Fin}\left(d\right)\to \mathrm{Fin}\left(3\right))\right))\right)$$

Formalization. D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.admissibleWords (✓ std3).

Citation. Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Commentary.

All matching words of length d are filtered by the connected-block condition. For positive depth their number is 3 times 2 to the power d-1.

Definition 1.7 (The uniform circuit expectation).

$$\forall q\in \mathbb{N},\forall d\in \mathbb{N},\mathrm{vecPhi}\left(q,d\right)=(((Finset.card\left(\mathrm{admissibleWords}\left(d\right)\right):\mathbb{C}){)}^{-1})\cdot (\sum _{w\in \mathrm{admissibleWords}\left(d\right)}\mathrm{wordMoment}\left(q,d,w\right))$$

Formalization. D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.vecPhi (✓ std3).

Citation. Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Commentary.

Section 2.1 (page 3) gives vec(Phi_epsilon) = E[U* tensor U* tensor U tensor U]. Here the expectation is the uniform average of the circuit-ordered layer-moment products over all admissible matching words; independent gates have already been integrated inside each layer. The reciprocal cardinality is taken in the complex numbers.

Definition 1.8 (Four-site permutation vectors).

$$\forall q\in \mathbb{N},\forall b\in (\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)),\forall r\in (((\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right))\times (\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right)))\times ((\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right))\times (\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(q\right)))),\mathrm{permutationVector}\left(q,b\right)\left(r\right)=\prod _{s:\mathrm{Fin}\left(4\right)}(\mathrm{sitePermutation}\left(q,b\left(s\right),((r.1.1\left(s\right),r.1.2\left(s\right)),(r.2.1\left(s\right),r.2.2\left(s\right)))\right))$$

Formalization. D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.permutationVector (✓ std3).

Citation. Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Commentary.

The four-site physical vector is the tensor product of the normalized one-site identity or swap vectors. No orthonormal coefficient basis is substituted for these physical vectors.

Definition 1.9 (The antisymmetric physical vector).

$$\forall q\in \mathbb{N},\mathrm{physicalWitness}\left(q\right)=\mathrm{permutationVector}\left(q,![0,1,0,1]\right)-\mathrm{permutationVector}\left(q,![0,1,1,0]\right)-\mathrm{permutationVector}\left(q,![1,0,0,1]\right)+\mathrm{permutationVector}\left(q,![1,0,1,0]\right)$$

Formalization. D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.physicalWitness (✓ std3).

Source. Repository-derived.

Acknowledgement. Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Commentary.

The vector is (|01>-|10>) on sites 0,1 tensored with (|01>-|10>) on sites 2,3, expressed in the nonorthogonal physical identity/swap vectors. Its squared norm is 4(1-q^(-2)) squared, which is positive for q at least two.

Definition 1.10 (The even-depth question).

$$claim=\left(\forall q\in \mathbb{N},(2\le q)\Rightarrow (\forall d\in \mathbb{N},(2\le d)\Rightarrow ((\mathrm{Even}\left(d\right))\Rightarrow (\neg Matrix.PosSemidef\left(\mathrm{vecPhi}\left(q,d\right)\right))))\right)$$

Formalization. D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Commentary.

Section 5.2 (page 12): “Like the brickwork, this architecture can be shown to have a PSD vectorization if the depth is odd. It is unclear if the vectorization is PSD at even depths.” The displayed claim is the negative answer on four sites: all q at least two and all even d at least two fail Matrix.PosSemidef.

Theorem 1.11 (Failure at every even depth).

$$\forall q\in \mathbb{N},(2\le q)\Rightarrow (\forall d\in \mathbb{N},(2\le d)\Rightarrow ((\mathrm{Even}\left(d\right))\Rightarrow (\neg Matrix.PosSemidef\left(\mathrm{vecPhi}\left(q,d\right)\right))))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.result (✓ std3). ∎

Resolves. Problems/belkin-allen-clark-2025-permuted-brickwork-even-depth (refuted) by D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.result.

Source. Repository-derived.

Acknowledgement. Daniel Belkin, James Allen, Bryan K. Clark (2025). Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits. URL: https://arxiv.org/abs/2510.23726v2.

Commentary.

Put a=q/(q squared + 1) and h=(q to the fourth + 1)/(q squared + 1) squared. The sum of the three layer projections has eigenvalue h on the physical vector. The unnormalized no-repeat word sum R obeys R_(d+1)=(S-I)R_d, while the word count is 3 times 2 to the power d-1. The average therefore has eigenvalue (h/3)(-a squared) to the power d-1 on this vector. Multiplying by its strictly positive squared norm gives a negative real quadratic form at every even depth. At q=2,d=2 this form is -51/625. A positive semidefinite matrix must have nonnegative quadratic forms, giving the contradiction.

References

• Truth anchor: D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.NoRepeat
• Truth anchor: D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.admissibleWords
• Truth anchor: D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.claim
• Truth anchor: D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.layerMoment
• Truth anchor: D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.matching
• Truth anchor: D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.permutationVector
• Truth anchor: D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.physicalWitness
• Truth anchor: D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.restrictReplica
• Truth anchor: D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.result
• Truth anchor: D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.vecPhi
• Truth anchor: D5/S3/Quantum/RandomCircuits/PermutedBrickworkEvenRefutation.wordMoment
• Dependency: D5/S3/Quantum/RandomCircuits/HaarTwoCopyTwirl