1. 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

2. 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

3. 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:

4. 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

5. 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

6. ? 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

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

8. 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?

9. 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

10. 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:

11. 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:

12. Transitions  history statesU1 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:

13. 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

14. 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

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

16. 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 )
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!!

17. “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

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

19. 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

20. 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

21. ? 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

22. 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?