SLIDE 1 Complexity Classification of Commuting Two-qubit Hamiltonians
Adam Bouland
Based on joint work with Laura Mančinska and Xue (Lucy) Zhang arXiv: 1602.04145 ECCC:TR16-039
SLIDE 2 Establishing Quantum Advantage
– Factoring – Approximating Jones polynomial
Evidence not in BPP:
- No known classical algorithm
Difficult to implement
SLIDE 3 Establishing Quantum Advantage
– Boson Sampling [Aaronson Arkhipov] – IQP [Bremner Jozsa Montanaro Shepherd] – Many others [Knill LaFlamme]
[Morimae Fuji Fitzsimons][Fefferman Umans]…
Evidence not in samp-BPP:
- Not in samp-BPP unless PH collapses
Possibly easier to implement
This work: Classify when you get quantum supremacy for sampling
SLIDE 4
Model Gate set = Unitary Hamiltonian
SLIDE 5
Model
Goal: Classify H according the power of poly- sized quantum circuits with e^iHt gates. Which H give you advantage over classical computation?
Gate set =
SLIDE 6
We classify the power of commuting 2-qubit Hamiltonians
=
SLIDE 7 Main Result: Dichotomy + Classification
For any 2-qubit commuting H:
- If H generates entanglement, then it
allows to you perform hard sampling problems
- Otherwise, H is efficiently classically
simulable
SLIDE 8 Hard Sampling Task
D
Hard to sample from classically
SLIDE 9
Hard to Sample
There does not exist samp-BPP algorithm M satisfying Assumption: The polynomial hierarchy doesn’t collapse
SLIDE 10
Relation to prior work
Previously: Knew some commuting H allow you to perform difficult sampling tasks This work: All commuting H (other than non- entangling ones) allow you to perform difficult sampling tasks
SLIDE 11 Relation to prior work
Generic Hamiltonians are universal [Lloyd ‘95] Some commuting Hamiltonians let you do hard sampling tasks [Bremner Jozsa Shepherd ‘10] Generic commuting Hamiltonians let you do hard sampling tasks [This work]
SLIDE 12
Relation to prior work
SLIDE 13
Why this matters
Easier to error correct [Aliferis et al. ‘09]
SLIDE 14
Proof Outline
Technique: Show postselected circuits with H are universal for BQP hardness of sampling by known techniques [Bremner Jozsa Shepherd, Aaronson Arkhipov]
SLIDE 15 Proof Outline
Commuting Hamiltonians PostBQP=PP BPP Postselection PostBPP BQP Postselection
SLIDE 16 Goal: Postselected Universality
- 1-qubit gates + any entangling Hamiltonian is
universal [Dodd et al. ‘02, Bremner et al. ‘02]
To complete proof: Get all 1-qubit gates under postselection
SLIDE 17
Goal: 1 qubit gates
If H is commuting, then H=
SLIDE 18
Goal: 1 qubit gates Postselection gadget:
SLIDE 19 Goal: 1 qubit gates
Suffices to show can perform all 1-qubit
- perations using products of L(t)’s
S is a group
SLIDE 20 Goal: 1 qubit gates
Suffices to show can perform all 1-qubit
- perations using products of L(t)’s
S is a group
Inverses?
SLIDE 21
Goal: 1 qubit gates
How do you invert postselection?
SLIDE 22
Goal: 1 qubit gates
No Solovay-Kitaev Theorem without inverses
SLIDE 23
Goal: 1 qubit gates
=
SLIDE 24
Goal: 1 qubit gates
L(t)’s (and their inverses) form a group & densely generate SL(2,C)
SLIDE 25
Last case
This works for all entangling H except Prior work: Hard because can embed permanents in output distribution
SLIDE 26 Open problems
- Complete the classification!
- Extend hardness to L1 error
- Classify commuting gate sets
SLIDE 27
Thanks! Questions?
