1. Fernando G.S.L. Brandão MSR -> Caltech Faculty Summit 2016
Quantum Speed-ups for Semidefinite Programming
Fernando G.S.L. Brandão
MSR -> Caltech
Faculty Summit 2016
based on joint work with
Krysta Svore
MSR

2. Quantum Algorithms Exponential speed-ups:
Simulate quantum physics, factor big numbers (Shor’s algorithm), …,
Polynomial Speed-ups:
Searching (Grover’s algorithm: N1/2 vs O(N)), ...
Heuristics:
Quantum annealing (adiabatic algorithm), machine learning, ...

3. Solving Semidefinite Programming belongs here
Quantum Algorithms
Exponential speed-ups:
Simulate quantum physics, factor big numbers (Shor’s algorithm), …,
Polynomial Speed-ups:
Searching (Grover’s algorithm: N1/2 vs O(N)), ...
Heuristics:
Quantum annealing (adiabatic algorithm), machine learning, ...
This Talk:
Solving Semidefinite Programming belongs here

4. Semidefinite Programming
… is an important class of convex optimization problems
Input: n x n, r-sparse matrices C, A1, ..., Am and numbers b1, ..., bm
Output: X
Some Applications: operations research (location probems, scheduing, ...), bioengineering (flux balance analysis, ...), approximating NP-hard problems (max-cut, ...), field theory (conformal bootstraping), ...
Algorithms Interior points: O((m2nr + mn2)log(1/ε))
Multilicative Weights: O((mnr/ε2))
Lower bound: No faster than Ω(nm), for constant ε and r

5. Semidefinite Programming
… is an important class of convex optimization problems
Input: n x n, r-sparse matrices C, A1, ..., Am and numbers b1, ..., bm
Output: X
Some Applications: operations research (location probems, scheduling, ...), bioengineering (flux balance analysis, ...), approximating NP-hard problems (max-cut, ...), ...
Algorithms Interior points: O((m2nr + mn2)log(1/ε))
Multilicative Weights: O((mnr/ε2))
Lower bound: No faster than Ω(nm), for constant ε and r

6. Semidefinite Programming
… is an important class of convex optimization problems
Input: n x n, r-sparse matrices C, A1, ..., Am and numbers b1, ..., bm
Output: X
Some Applications: operations research (location probems, scheduling, ...), bioengineering (flux balance analysis, ...), approximating NP-hard problems (max-cut, ...), ...
Algorithms Interior points: O((m2nr + mn2)log(1/ε))
Multiplicative Weights: O((mnr (⍵R)/ε2))
Lower bound: No faster than Ω(nm), for constant ε and r
“width”
“size of solution”

7. Semidefinite Programming
… is an important class of convex optimization problems
Input: n x n, r-sparse matrices C, A1, ..., Am and numbers b1, ..., bm
Output: X
Some Applications: operations research (location probems, scheduling, ...), bioengineering (flux balance analysis, ...), approximating NP-hard problems (max-cut, ...), ...
Algorithms Interior points: O((m2nr + mn2)log(1/ε))
Multiplicative Weights: O((mnr (⍵R)/ε2))
Lower bound: No faster than Ω(nm), for constant ε, r, ⍵, R

8. Semidefinite Programming
Normalization:
We assume:
Reduction optimization to decision: or

9. SDP Duality
Primal:
Dual:

10. Quantum Algorithm for SDP
thm There is a quantum algorithm for solving SDPs running in time Õ(n1/2 m1/2 r δ-2 R2)

11. Quantum Algorithm for SDP
thm There is a quantum algorithm for solving SDPs running in time Õ(n1/2 m1/2 r δ-2 R2)
Input: n x n matrices r-sparse C, A1, ..., Am and numbers b1, ..., bm
Output: Samples from y/||y||2 and ||y||2
Q. Samples from X/tr(X) and tr(X)
Primal:
Dual: or

12. Quantum Algorithm for SDP
thm There is a quantum algorithm for solving SDPs running in time Õ(n1/2 m1/2 r δ-2 R2)
Input: n x n matrices r-sparse C, A1, ..., Am and numbers b1, ..., bm
Output: Samples from y/||y||2 and ||y||2
Q. Samples from X/tr(X) and tr(X)
Oracle Model: We assume there’s an oracle that outputs a chosen non-zero entry of C, A1, ..., Am at unit cost:
choice of Aj row k l non-zero element

13. Quantum Algorithm for SDP
thm There is a quantum algorithm for solving SDPs running in time Õ(n1/2 m1/2 r δ-2 R2)
Classical lower bound At least time Ω(max(n, m) for r, δ, R = Ω(1). Reduction from Search
Quantum lower bound At least time Ω(max(n1/2, m1/2) for r, δ, R = Ω(1). Reduction from Search
Ex.
Ω(max(m)): bi = 1, C = I
Aj = I for all j (obj = 1)
Aj = 2I for random j in [m] and Ak = I for k ≠ j (obj = 1/2)

14. Quantum Algorithm for SDP
thm There is a quantum algorithm for solving SDPs running in time Õ(n1/2 m1/2 r δ-2 R2)
thm 2 There is a quantum algorithm for solving SDPs running in time Õ(TGibbs m1/2 r δ-2 R2)
If Gibbs states can be prepared quickly (e.g. by quantum metropolis), larger speed-ups possible

15. Quantum Algorithm for SDP
The quantum algorithm is based on a classical algorithm for SDP due to Arora and Kale (2007) based on the multiplicative weight method
Let’s review their method

16. The Oracle
Searches for a vector y s.t. i)
ii)
Dual:

17. Width of SDP

18. Arora-Kale Algorithm 1. 2. 3. 4.

19. Why Arora-Kale works? Since Must check is feasible From Oracle,
We need:

20. Matrix Multiplicative Weight
MMW (Arora, Kale ‘07) Given n x n matrices Mt and ε < ½,
with and
λn : min eigenvalue
2-player zero-sum game interpretation:
- Player A chooses density matrix Xt
- Player B chooses matrix 0 < Mt<I
Pay-off: tr(Xt Mt)
“Xt = ⍴t strategy almost as good as global strategy”

21. Arora-Kale Algorithm 1. 2. 3. 4.

22. Arora-Kale Algorithm 1. 2. 3. 4.
How to implement the Oracle?

23. Implementing Oracle by Gibbs Sampling
Searches for a vector y s.t. i)
ii)

24. Implementing Oracle by Gibbs Sampling
Searches for (non-normalized) probability distribution y satisfying two linear constraints:
claim: We can take Y to be Gibbs: There are constants N, x, y s.t.

25. Jaynes’ Principle Then there is a Gibbs state of the form
(Jaynes 57) Let ⍴ be a quantum state s.t.
Then there is a Gibbs state of the form
with same expectation values.
Drawback: no control over size of the λi’s.

26. Finitary Jaynes’ Principle
(Lee, Raghavendra, Steurer ‘15) Let ⍴ s.t.
Then there is a
with
s.t.
(Note: Used to prove limitations of SDPs for approximating constraints satisfaction problems)

27. Implementing Oracle by Gibbs Sampling
Claim There is a Y of the form
with x, y < log(n)/ε and N < α s.t.
N < α:

28. Implementing Oracle by Gibbs Sampling
Claim There is a Y of the form
with x, y < log(n)/ε and N < α s.t.
Can implement oracle by exhaustive searching over x, y, N for a Gibbs distribution satisfying constraints above
(only α log2(n)/ε3 different triples needed to be checked)

29. Arora-Kale Algorithm 1. 2. 3. 4. Using Gibbs Sampling
Again, it’s Gibbs Sampling

30. Quantum Speed-ups for Gibbs Sampling
(Poulin, Wocjan ’09, Chowdhury, Somma ‘16) Given a r-sparse D x D Hamiltonian H, one can prepare exp(H)/tr(…) to accuracy ε on a quantum computer with O(D1/2 r ||H|| polylog(1/ ε)) calls to an oracle for the entries of H
Based on amplitude amplification

31. Quantizing Arora-Kale1. 2. 3. 4.
Use Gibbs Sampling
Again, Gibbs Sampling

32. Quantizing Arora-Kale
Problem: Mt is not sparse.
Solution: Sparsify it using operator Chernoff bound 1. 2. 3. 4.
Using Gibbs Sampling
Again, Gibbs Sampling

33. Sparsification
Suppose Z1, …, Zk are independent n x n Hermitian matrices s.t. E(Zi)=0, ||Zi||<λ. Then
Applying to
Sampling j1, …, jk from with k = O(log(n)/ε2)

34. Quantizing Arora-Kale1. 2. 3. 4.
Use Gibbs sampling
Sparsify by random sampling
Again, Gibbs sampling

35. Quantizing Arora-Kale1. 2. 3. 4.
Use Gibbs sampling
Takes Õ(n1/2r) time on a quantum computer
Sparsify by random sampling
Again, Gibbs sampling

36. Quantizing Arora-Kale1. 2. 3. 4.
Use Gibbs sampling
Needs to prepare: exp(∑i hi |i><i|)/tr(…) with hi = x tr(Ai ⍴t) + y bi
Can do quantumly with Õ(m1/2) calls to an oracle to h.
Each call to h consists in estimating the value of tr(Ai ⍴t).
To estimate needs to prepare copies of ⍴t, which costs Õ(n1/2r)
Overall: Õ(n1/2m1/2r)
Sparsify by random sampling

37. Conclusion and Open Problems
We showed quantum computers can speed-up the solution of SPDs
One application is to solve the Goesman-Williamson SDP for Max-Cut in time Õ(|V|). Are there more applications?
- Can we improve the parameters?
Can we find (interesting) instances where there are superpolynomial speed-ups?
Quantum speed-up for interior-point methods?
Thanks!