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."

? = 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

Soap bubble

Why we believe PNP

"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)

\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

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

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

Quantum supremacy

By Ninnat Dangniam

Quantum supremacy

QTFT (Quantum Technology Foundation of Thailand) theory webinar

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