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 

? = 239593

P vs NP

Does the ability to quickly recognize a correct answer imply the ability to quickly solve the problem?

263 \times 911 = 239593

P vs NP

Decision problem solvable in polynomial time

  • Asymptotic
  • Worst-case 

(Big O notation)

Time complexity, Wikipedia

Not \(10000 = 2 \times \cdots\)

  • 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 \(ab\ge c\)?

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: CK, Akshita Sahni

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
\implies
\implies

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: \(\mathrm{A}^O\)

\(\mathrm{P}^O\): Poly-time solvable with oracle \(O\) 

\(\mathrm{NP}^O\): Poly-time verifiable with oracle \(O\)

Polynomial hierarchy

\(\mathrm{P}^O\): Poly-time solvable with oracle \(O\) 

\(\mathrm{NP}^O\): Poly-time verifiable with oracle \(O\)

\mathrm{NP} \subseteq \mathrm{NP^{NP}} \subseteq \mathrm{NP^{NP^{NP}}} \subseteq \cdots
\mathrm{P^P = P}

"Self-low" (physical) complexity class

\Sigma_{k+1} = \mathrm{NP}^{\Sigma_k}
=\Sigma_0
\Sigma_1
\Sigma_2
\Sigma_3
=
=
=

Polynomial hierarchy \(\mathrm{PH}\)

Believed not to collapse for a similar reason that P≠NP

Notation: \(\mathrm{A}^O\)

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

 

\implies

(in certain tasks)

Sandia mountains, Albuquerque 2018

PH collapses to the 3rd level

(Does not imply P=NP)

\implies

(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

\(\mathrm{BPP} \subseteq \Sigma_2\) (Sipser–Gács–Lautemann 1983)

\(\#\mathrm{P}\) counts the number of solutions to an NP problem (\(\mathrm{NP} \subseteq \mathrm{P^{\# P}}\))

US50K

How many routes have length \(\le k\) cities?

\(\mathrm{PH \subseteq P^{\# P}}\) (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

p_z = |\langle z|U|0\rangle^{\otimes n}|^2
\textrm{Pr(correct)} = \displaystyle{ \frac{\# \textrm{correct answers}}{2^n} }

Caution: Quantum computers are not known to be able to solve  \(\mathrm{NP}\)-complete (let alone \(\#\mathrm{P}\)-complete) problems efficiently!

The ability to sample from a hard-to-calculate distribution does not give a quick way to solve a \(\#\mathrm{P}\)-complete problem...

Unless you also have an NP oracle (even then it only almost works)

  • Argue that "good" approximations to most output probabilities \(p_z\) of a certain quantum device let you solve a  \(\mathrm{\# P}\)-complete problem (average-case complexity; currently lacking)
  • Prove that a \(\mathrm{BPP^{NP}}\) machine that can sample \(p_z\) can output good approximations to most \(p_z\)

Quantum supremacy argument

(very rough sketch)

\mathrm{P^{\# P} \subseteq BPP^{NP}}
\implies\Sigma_3 = \mathrm{PH}
\subseteq \Sigma_3
\mathrm{PH \subseteq}

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