A property-T circulant with universal perfect state transfer

Abstract

The circulant Circ(0, a, conj(a)) with a = (-4 sqrt(3) + i)/7 is a Hermitian circulant on three vertices whose nonzero entries have modulus 1 and whose continuous-time quantum walk has perfect state transfer between every pair of vertices, yet it is switching equivalent neither to K_2 nor to Circ(0, -i, i). This refutes the conjecture of E. Connelly, N. Grammel, M. Kraut, L. Serazo and C. Tamon (arXiv:1701.04145) that these two are the only circulants with property T and universal perfect state transfer.

Definition 1.1 (Property T).

$$\mathrm{PropertyT}\left(A\right)\iff (\forall jk,(A\left(j,k\right)\ne 0)\Rightarrow \Vert A\left(j,k\right)\Vert =1)$$

Formalization. D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.PropertyT (✓ std3).

Citation. Erin Connelly, Nathaniel Grammel, Michael Kraut, Luis Serazo, Christino Tamon (2017). Universality in perfect state transfer. DOI: 10.1016/j.laa.2017.06.015. URL: https://arxiv.org/abs/1701.04145v2.

Commentary.

A matrix A has property T when every nonzero entry of A has modulus 1. For the adjacency matrix of a graph this says that the graph is a complex unit gain graph.

Definition 1.2 (Universal perfect state transfer).

$$\mathrm{UPST}\left(A\right)\iff (\forall uv,\exists t\in \mathbb{R},\Vert \mathrm{hamiltonianPropagator}\left(A,t,v,u\right)\Vert =1)$$

Formalization. D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.UPST (✓ std3).

Citation. Erin Connelly, Nathaniel Grammel, Michael Kraut, Luis Serazo, Christino Tamon (2017). Universality in perfect state transfer. DOI: 10.1016/j.laa.2017.06.015. URL: https://arxiv.org/abs/1701.04145v2.

Commentary.

The continuous-time quantum walk of a Hermitian matrix A is U(t) = exp(-i t A). There is perfect state transfer from u to v at time t when the (v, u) entry of U(t) has modulus 1, and A has universal perfect state transfer when this happens, at some real time, for all vertices u and v.

Definition 1.3 (Switching equivalence).

$$\mathrm{SwitchingEquivalent}\left(A,B\right)\iff (\exists \sigma ,\exists d,(\forall i,d\left(i\right)\ne 0)\wedge (\mathrm{perm}\left(\sigma \right)\mathrm{diag}\left(d\right)A=B\mathrm{perm}\left(\sigma \right)\mathrm{diag}\left(d\right)))$$

Formalization. D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.SwitchingEquivalent (✓ std3).

Citation. Erin Connelly, Nathaniel Grammel, Michael Kraut, Luis Serazo, Christino Tamon (2017). Universality in perfect state transfer. DOI: 10.1016/j.laa.2017.06.015. URL: https://arxiv.org/abs/1701.04145v2.

Commentary.

A monomial matrix is the product of a permutation matrix and an invertible diagonal matrix, and two matrices A and B are switching equivalent when M A = B M for some monomial matrix M. Here M is the permutation matrix of a bijection sigma between the vertex sets times the diagonal matrix of a vector d with no zero entry.

Definition 1.4 (The complete graph on two vertices).

$$K2=\mathrm{Circ}\left(0,1\right)$$

Formalization. D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.K2 (✓ std3).

Citation. Erin Connelly, Nathaniel Grammel, Michael Kraut, Luis Serazo, Christino Tamon (2017). Universality in perfect state transfer. DOI: 10.1016/j.laa.2017.06.015. URL: https://arxiv.org/abs/1701.04145v2.

Commentary.

The circulant Circ(0, 1) on Z/2Z, the adjacency matrix of K_2. Here Circ(a) is the circulant matrix with entries Circ(a)(jk) = a(k - j), the transpose of the circulant matrix of Mathlib.

Definition 1.5 (The oriented triangle).

$$\mathrm{orientedTriangle}=\mathrm{Circ}\left(0,-i,i\right)$$

Formalization. D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.orientedTriangle (✓ std3).

Citation. Erin Connelly, Nathaniel Grammel, Michael Kraut, Luis Serazo, Christino Tamon (2017). Universality in perfect state transfer. DOI: 10.1016/j.laa.2017.06.015. URL: https://arxiv.org/abs/1701.04145v2.

Commentary.

The circulant Circ(0, -i, i) on Z/3Z, the oriented triangle, which has universal perfect state transfer.

Definition 1.6 (The conjecture).

$$claim\iff (\forall n,(n\ge 2)\Rightarrow \forall a:\mathrm{ZMod}\left(n\right)\to \mathbb{C},(a\left(0\right)=0)\Rightarrow \left((\mathrm{IsHermitian}\left(\mathrm{Circ}\left(a\right)\right))\Rightarrow \left((\mathrm{PropertyT}\left(\mathrm{Circ}\left(a\right)\right))\Rightarrow \left((\mathrm{UPST}\left(\mathrm{Circ}\left(a\right)\right))\Rightarrow \left(\mathrm{SwitchingEquivalent}\left(\mathrm{Circ}\left(a\right),K2\right)\vee \mathrm{SwitchingEquivalent}\left(\mathrm{Circ}\left(a\right),\mathrm{orientedTriangle}\right)\right)\right)\right)\right))$$

Formalization. D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.claim (✓ std3).

Citation. Erin Connelly, Nathaniel Grammel, Michael Kraut, Luis Serazo, Christino Tamon (2017). Universality in perfect state transfer. DOI: 10.1016/j.laa.2017.06.015. URL: https://arxiv.org/abs/1701.04145v2.

Commentary.

The conjecture of the paper, read together with its introduction, which names K_2 and Circ(0, -i, i) as the only known examples and conjectures that this set is unique: every Hermitian circulant Circ(a) on Z/nZ with n at least 2, without loops (a_0 = 0), with property T and universal perfect state transfer is switching equivalent to K_2 or to Circ(0, -i, i).

Theorem 1.7 (A counterexample on three vertices).

$$\neg claim$$

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

Resolves. Problems/connelly-2017-property-t-universal-state-transfer (refuted) by D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.result.

Source. Repository-derived.

Acknowledgement. Erin Connelly, Nathaniel Grammel, Michael Kraut, Luis Serazo, Christino Tamon (2017). Universality in perfect state transfer. DOI: 10.1016/j.laa.2017.06.015. URL: https://arxiv.org/abs/1701.04145v2.

Commentary.

Let a = (-4 sqrt(3) + i)/7, so that |a| = 1, and let A = Circ(0, a, conj(a)) on Z/3Z. Then A is Hermitian, has no loops and has property T. The Fourier matrix V with entries omega^(km), where omega = exp(2 pi i/3), satisfies A V = V diag(mu) with eigenvalues mu = (sqrt(3)/7)(-8, 3, 5). At t_1 = 14 sqrt(3) pi/9 the numbers -i t_1 mu_m are 2 pi i/3 times 2, 0 and 1 modulo 2 pi i, so exp(-i t_1 A) = V diag(omega^2, 1, omega) V^(-1) = omega^2 P, where P is the cyclic permutation matrix with P_(jk) = 1 exactly when k = j + 1. Hence U(t_1) = omega^2 P, U(2 t_1) = U(t_1)^2 = omega^4 P^2 and U(0) = I, and every entry of these matrices on the relevant pair of vertices has modulus 1, which gives universal perfect state transfer. The vertex sets of A and K_2 have 3 and 2 elements, so there is no bijection between them. A monomial matrix M has nonzero determinant, so M A = B M gives det A = det B; but det A = a^3 + conj(a)^3 = -360 sqrt(3)/343, while det Circ(0, -i, i) = 0. So A is switching equivalent to neither.

References

• Truth anchor: D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.K2
• Truth anchor: D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.PropertyT
• Truth anchor: D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.SwitchingEquivalent
• Truth anchor: D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.UPST
• Truth anchor: D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.claim
• Truth anchor: D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.orientedTriangle
• Truth anchor: D5/S3/Quantum/Dynamics/PropertyTCirculantUniversalTransfer.result
• Dependency: D5/S3/Quantum/Dynamics/ProjectionProbabilityFlow