Towards Quantum Supremacy in Near-Term Quantum Computers
QTFT Webinar, 7 Sep 2019
Supanat Thanasilp
Ninnat Dangniam
Department of Physics, Fudan University
Centre for Quantum Technologies, National University of Singapore
Shor's algorithm is not a proof of quantum supremacy
"It's important to keep in mind that there is no conceptual reason why factoring should be difficult. By contrast, one can make a compelling case for why P is different from NP, but factoring is almost undoubtedly not NP-hard. People have made immense progress in developing factoring algorithms, and there's no reason to think we've hit a fundamental barrier."
Henry Cohn, Factoring may be easier than you think
P vs NP
Does the ability to quickly recognize a correct answer imply the ability to quickly solve the problem?
P vs NP
Decision problem solvable in polynomial time
- Asymptotic
- Worst-case
(Big O notation)
Time complexity, Wikipedia
Not 10000=2×⋯
- Multiplication
- Primality testing
- Linear programming
- Edit distance
- Stable marriage
Examples of P problems
Decision problem solvable in polynomial time
Decision version: given integers a,b, is ab≥c?
P vs NP
Decision problem verifiable in polynomial time
Factoring
A way to quickly solve an NP-hard problem would give a way to quickly solve every NP problem
A problem is NP-complete if it is NP-hard and belongs to NP
"Hardest NP problems"
- Traveling salesman
- Job scheduling
- Packing
- Circuit design (SAT)
- Graph coloring
"NP-intermediate"
P vs NP
NP-hard problems in physics
3D Ising model
Cr: iStock/sergunt
Protein folding
Cr: Mateus Schmidt
Soap bubble
Why we believe P≠NP
"If P=NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in “creative leaps,” no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss; everyone who could recognize a good investment strategy would be Warren Buffett."
Scott Aaronson, Reasons to believe
(Disowned in 2014 for possible confusion with the human vs AI issue)
"ถ้า P = NP ลอกการบ้านก็ไม่เร็วไปกว่าทำเอง"
- Separate P and NP (although more than a hundred have tried)
- Separate classical and quantum computing
- Show that the collapse of one implies the collapse of the other
Separation
Collapse
No one knows how to...
Possible using weaker "collapse of the polynomial hierarchy"!
Oracle
An oracle is a black box that instantly solves problem O
Example: NP-complete problem O gives an NP oracle
Does 100001 has a prime factor less than 13?
Yes
Wow thanks! Now I can steal my mom's credit card
Notation: AO
PO: Poly-time solvable with oracle O
NPO: Poly-time verifiable with oracle O
Polynomial hierarchy
PO: Poly-time solvable with oracle O
NPO: Poly-time verifiable with oracle O
"Self-low" (physical) complexity class
Polynomial hierarchy PH
Believed not to collapse for a similar reason that P≠NP
Notation: AO
Quantum supremacy argument
PH collapses to the 3rd level
(Does not imply P=NP)
Aaronson and Arkhipov, The Computational Complexity of Linear Optics, STOC 2011
Bremner, Jozsa and Shepherd, Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy, PRSA 2010
(in certain tasks)
Sandia mountains, Albuquerque 2018
PH collapses to the 3rd level
(Does not imply P=NP)
(in certain tasks)
Quantum supremacy argument
Aaronson and Arkhipov, The Computational Complexity of Linear Optics, STOC 2011
Bremner, Jozsa and Shepherd, Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy, PRSA 2010
Polynomial hierarchy & friends
Sensible model of probabilistic computation
BPP⊆Σ2 (Sipser–Gács–Lautemann 1983)
#P counts the number of solutions to an NP problem (NP⊆P#P)
How many routes have length ≤k cities?
PH⊆P#P (Toda 1989)
Sampling problems
- Tasks that show quantum supremacy need not be useful
- Quantum computers always output probability samples
- Exactly calculating output probabilities from a uniform input distribution is the same as counting
Galton board
Caution: Quantum computers are not known to be able to solve NP-complete (let alone #P-complete) problems efficiently!
The ability to sample from a hard-to-calculate distribution does not give a quick way to solve a #P-complete problem...
Unless you also have an NP oracle (even then it only almost works)
- Argue that "good" approximations to most output probabilities pz of a certain quantum device let you solve a #P-complete problem (average-case complexity; currently lacking)
- Prove that a BPPNP machine that can sample pz can output good approximations to most pz
Quantum supremacy argument
(very rough sketch)
Sipser–Gács–Lautemann
Toda
Collapses to the 3rd level
Summary
- "Quantum speedups" alone are not proofs of "quantum supremacy"
- We believe, but do not know how to prove or disprove P≠NP
- We do not have a quantum supremacy argument based on P≠NP
- We do have a quantum supremacy argument based on the implausibility of the PH collapse
- With 50-100 noisy qubits, we may be able to see an experimental test of quantum supremacy
Further readings
Computational complexity
- Arora and Barak, Computational Complexity: A Modern Approach, CUP 2009
- Aaronson, Quantum Computing Since Democritus, CUP 2013 [Course lectures]
Quantum supremacy
- Harrow and Montanaro, Quantum Computational Supremacy, Nature 2017 [Free version]
- Lund, Bremner and Ralph, Quantum sampling problems, BosonSampling and quantum supremacy, npj Quantum Information 2017
- Bouland, Fefferman, Nirkhe and Vazirani, Quantum supremacy and the complexity of random circuit sampling (preprint), 2018
