Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing Circuit Relations for Real Stabilizers: Towards TOF + H Cole Comfort University of Calgary May 23, 2019 1/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing Outline 1 Graphical calculi for circuits 2 Classical reversible computing 3 Quantum computing 4 Classical reversible and quantum computing 2/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing Graphical calculi and completeness The graphical calculus for PROPs models circuit computation. Finite presentations of different fragments of computing are studied. Complete presentation is a strict y-symmetric monoidal faithful functor. For quantum computing: ZX-calculus, ZH-calculus, ZW-calculus. For reversible computing: CNOT and TOF. 3/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing The category CNOT Consider the PROP generated by cnot, j1i, h1j: The controlled not gate, cnot, takes bits: jb1; b2i 7! jb1; b1 ⊕ b2i cnot is drawn as: j1i is preparing 1 and h1j is postselecting 1: The not gate and j0i, h0j are derived: := := := This category has a finite, complete presentation in terms of circuit identities, CNOT [CCS18]: 4/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing The identities of CNOT 5/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing The category TOF Consider the PROP generated by tof, j1i, h1j: The Toffoli gate, tof, takes bits: jb1; b2; b3i 7! jb1; b1 · b2 ⊕ b1i tof is drawn as: j1i is preparing 1 and h1j is postselecting 1: The cnot gate is derived: := This category has a finite, complete presentation, TOF [CC19]: 6/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing The identities of TOF 7/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing The identities of TOF 8/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing Both CNOT and TOF have concrete equivalent categories. In particular they are discrete inverse categories. That means that they have a total copying map generated by: := and := 9/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing Frobenius algebras A Frobenius algebra is a monoid-comonoid pair: = = = = = = Satisfying the Frobenius law: = = 10/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing Frobenius algebras A Frobenius algebra is commutative if: = = And special if: = Connected components of Frobenius algebras can be uniquely represented by spiders: . = ··· = 11/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing Frobenius algebras are bases in FdHilb Theorem ([CPV13]) Orthonormal bases fjiigi2B in FdHilb are in one-to-one correspondence with special, commutative y-Frobenius algebras: X X X X jii jiihi; ij hij ji; iihij i2B i2B i2B i2B Therefore, we can consider the Frobenius algebras associated to the eigenbasis of quantum observables. For example, consider the Hermetian matrices: 0 1 1 0 X := Z := 1 0 0 −1 X and Z have spectra: p p X+ = j+i = 1= 2(j0i + j1i) X− = |−i = 1= 2(j0i − j1i) Z+ = j0i; Z− = j1i 12/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing ZXπ The phase-free ZX-calculus, ZXπ, [DP13] is the PROP generated by the Z Frobenius algebra and Hadamard gate: The Hadamard gate is a self-inverse change of basis matrix so that: X+H = Z+HX−H = Z−HZ+H = X+HZ−H = X−H The Frobenius algebra for X is therefore given by conjugation. := := := := 13/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing ZXπ ZXπ has a finite presentation: The first identity is that the axioms of a special y-Frobenius algebra hold for Z . The Frobenius algebras associated to the Z and X observables are strongly complimentary. They form a Hopf algebra up to an invertible scalar: This corresponds to the bases being mutually unbiased [CD11]. 14/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H 15/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing CNOT + H p Consider the extension of CNOT with the Hadamard gate (and 2): 16/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing ZXπ ! CNOT + H Consider G : ZXπ ! CNOT + H, sending: p 2 7! 7! p 2 7! 7! 7! 7! 17/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing CNOT + H ! ZXπ Consider F : CNOT + H ! ZXπ, sending: 7! 7! 7! 7! π π p 2 7! 7! 18/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing CNOT is isomorphic to ZXπ Proposition F : CNOT + H ! ZXπ and G : ZXπ ! CNOT + H are y-preserving symmetric monoidal functors. Theorem F : CNOT + H ! ZXπ and G : ZXπ ! CNOT + H are inverses. This implies that CNOT + H is complete... Theorem ([DP13]) ZXπ is complete for real stabilizer circuits. p We can also remove the scalar 2 by being careful. 19/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing p p 2 2 7−!G = ··· = 20/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing Stabilizer circuits and universality There is a caveat: Theorem ([Got98]) Stabilizer circuits can be simulated in polynomial time on a classical probabilistic computer. However, Theorem ([Aha03]) The Toffoli and Hadamard gates, together are an approximately universal gate set for quantum computing. 21/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing Is there a presentation in terms of the Toffoli gate and H? ∼ But 6∼ 22/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing The Toffoli gate in ZXπ The Toffoli gate has the following representation in ZXπ [NW18]: π π = However, the Triangle has the following representation in CNOT + H [Vil18]: = p p 2 2 23/26 Cole Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing The ZH-calculus The controlled-Z gate can be represented with Toffoli gate and Hadamard: := In the ZH calculus controlled Z-gates are given by: . Axiom [H.F] of CNOT + H generalizes to Toffoli gates: [H.F’] = 24/26 Question: Is this enough to beCole complete? Comfort Circuit Relations for Real Stabilizers: Towards TOF + H Graphical calculi for circuits Classical reversible computing Quantum computing Classical reversible and quantum computing ReferencesI Dorit Aharonov. A simple proof that toffoli and hadamard are quantum universal. arXiv preprint quant-ph/0301040, 2003. Robin Cockett and Cole Comfort. The category TOF. Electronic Proceedings in Theoretical Computer Science, 287:67–84, 2019. Robin Cockett, Cole Comfort, and Priyaa Srinivasan. The category CNOT. Electronic Proceedings in Theoretical Computer Science, 266:258–293, 2018. Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13(4):043016, 2011. 25/26 Cole Comfort Circuit Relations for Real Stabilizers: