Hamiltonian quantum computer in one dimension Tzu-Chieh Wei C.N. Yang Institute for Theoretical Physics Department of Physics & Astronomy John C Liang Rumson-Fair High School  Stanford University Supported by AQIS, Taipei, 8/30/2016

Outline Introduction: Hamiltonian quantum computer 1D 3-local 5-state construction (not translation invariant) 1D 3-local 8-state construction (translation invariant) [will not have much time for this part] Summary

Quantum computation: circuit n qubits (initialized) round 1 round 2 round R of gates Measurement 3 steps: (1) Initialization, (2) Gate operations (3) Measurement Gates: a finite (universal) set of unitary transformations are sufficient 1-qubit gates: 2-qubit gate: Note 1-qubit gates are special case of 2-qubit gates:

Simulating circuit with Hamiltonian n qubits (initialized) round 1 round 2 round R of gates u v Measurement First Hamiltonian Quantum Computer by Feynman in 1984  Unary Clock: t=1: 00….001, t=2: 00….010, t=3: 00….100 (using σ’s to flip)  Ai applies to pair (u,v) of qubits (e.g. from circuit) clock  Each term is a 4-body (but not geometrically local) qubits u v

1D Hamiltonian Quantum Computer Can we do this with a one-dimensional Hamiltonian? (Feynman’s not 1D) Yes: but requires large local dimension for short-ranged, e.g. nearest-neighbor: Vollbrecht & Cirac: translation invariant, 30-state Kay: translation invariant, 31-state Nagaj & Wocjan: translation invariant, (1) 10-state (2) 20-state Aharonov et al. has one construction that (can be interpreted in terms of Hamiltonian QC) is non-translation invariant with 9-state Chase & Landahl: non-translation invariant, 8-state

? Locality k vs local dimension d 1D Local Hamiltonians (non-translationally invariant) 9 Aharonov et al. d 8 Chase & Landahl 7 ? 6 Local dimension 5 Not Universal This work 4 Implied by Chase & Landahl 3 2 1 2 3 4 5 6 Locality k

Outline Introduction: Hamiltonian quantum computer 1D 3-local 5-state construction (not translation invariant) 1D 3-local 8-state construction (translation invariant) Summary

Universal Circuit n qubits round 1 round 2 round R of gates How do we simulate such circuit on 1D chain? (How to modify Feynman’s construction?) u v clock qubits  Can we get rid of clock qubits?  How to apply different gates on the same pair of qubits?

Simulate circuit with 1D chain n qubits round 1 round 2 round R of gates Solution inspired by 1D QMA LHP: (1) Replace clock by pattern of symbols (2) Each gate is applied at specific & distinct location [Aharonov, Gottesman, Irani & Kempe ‘09] Example: n=3 qubits, R=2 rounds (initial state shown) 2 gates in one round [applied between neighboring qubits]  qubits need to be moved from block to block for next round of gates

Local Hilbert space & transition rules Two different types of sites: A: 5-state : two kinds of qubits : unborn/dead (1 dim) B: 5-state : spacer btwn qubits or unborn right/ left turn -around : movement & direction change Transition rules: 1: 1 (backward): 2: 6a: 3: boundary 6b: 4: NOT boundary 7a: 5a: 7b: 5b:

Transitions  history states [w qubits properly initialized in 0 or 1] U1 U2 U3 U4 (initial state) 6a 1: U1 1 2: U2 1 2 3: 3 4: 4 5a: 5a 5b: 4 6a: 5a 6b: 4 5b 7a: 6b 7b:

Transitions  history states U1 U2 U3 U4 6b 1: 7a 2: 7b 3 3: 4 4: 5a 5a: 5a 5b: 4 6a: 5b 6b: 6a U3 1 7a: U4 (final) 1 7b:

Computation at discrete time & history states Unique forward/backward transition (except at initial and final state) 6a 1 U1 1 U2 Transition rules uniquely connect history states (of computation) 2 3 5a But isn’t our computer run by continuous-time evolution?  Hamiltonian? 4 5b 6a 1 U3 U4 1

Transition rules  Yes, but can turn these rules into a Hamiltonian 1: 1 (backward): 2: 6a: 3: 6b: 4: 7a: 5a: 7b: 5b:  Yes, but can turn these rules into a Hamiltonian

Hamiltonian Constructed from the transition rules: In the basis of valid history (via transition rules):  Effective Hamiltonian:

Run your Hamiltonian computer [@|□+ □ + □ + □ + □.|O.O.O.O.O.|O.O.O.O.O.|O.O.O.O.O.|O.O.O.O.O.|O.O.O.O.O.] With qubits appropriately initialized (e.g. 00000) Quantum computation: evolve by Schrodinger equation via Hamiltonian Readout: measure in the “computational basis” Problems: But at what time t ? We want it to evolve to final state!  Naïve counting: probability 1/T to land onto the final state!!

“Raising” probability to finish goal Trick: set your training goal higher  pad a lot of identity gates I after desired rounds finished II I desired goal training goal  Then there is high probability of success in getting the correct outcome [.|O.O.O.O.O.|O.O.O.O.O.|O.O.O.O.O.|O.O. □ + □ + □ <|O+ □ + □.O.O.|O.O.O.O.O.] Remaining gates are identity  computation finished

Effective Hamiltonian: 1D “quantum walk” or tight-binding model has eigenvalues with eigenstates Starting at |0›, probability of arriving at |m› after time τ

Analysis for success probability Starting at |0›, probability of arriving at |m› after time τ Can show that [Nagaj & Wocjan, PRA 2008] Pad sufficient identity gates (e.g. 5 times as many) so that for m ≥ T/6, desired computation is done Readout: measure in the basis at random time A: B: e.g. [.|O.O.O.O.O.|O.O.O.O.O.|O.O.O.O.O.|O.O. □ + □ + □ <|O+ □ + □.O.O.|O.O.O.O.O.] Take  finite and high probability of success

So we have demonstrated a 1D 3-local 5-state (spin-2) Hamiltonian capable of universal QC  Classical simulation of such spin-2 Hamiltonian is BQP-complete

? What if translational invariance is imposed? 10 d 9 8 7 6 1D Local Hamiltonians (translationally invariant w.r.t. unit cells) 10 Nagaj & Wocjan d 9 ? 8 This work 7 6 Local dimension Not Universal 5 Implied by Nagaj & Wocjan 4 3 2 1 2 3 4 5 6 Locality k

Summary & open questions 1D 3-local 5-state (spin-2) Hamiltonian (not translationally invariant)  universal for Hamiltonian quantum computation 1D 3-local 8-state (spin-7/2) Hamiltonian (translationally invariant)  universal for Hamiltonian quantum computation Open: are the above results optimal? What about 1D 3-local QMA Hamiltonians? Minimum local dim?