Quantum advantage

Mahasarakham University, 26 Jan 2022

Ninnat Dangniam

The Institute for Fundamental Study (IF)

classical hardness in correlation games and random sampling

Two Views of Quantum Theory

Quantum theory as limitation

Uncertainty relation

Quantum theory as liberator

No objective reality

Faster algorithms

Unbreakable cryptography 

Engineering for quantum advantage

Many flavors of quantum advantage

  • Quantum-enhanced precision measurement
    • Caves (PRD 1981) proposed injecting "squeezed light" to improve gravitational wave detection sensitivity
    • 50% reduction of noise = 8 times the observable volume of the universe (Lough et al., PRL 2021)
  • Algorithmic quantum speedup
    • Shor's algorithm factorizes an integer \(N\) in \(\sim \log^3 N\) time instead of \(\sim\exp(\log^{1/3} N)\)
    • No large-scale demonstration
  • "Quantum supremacy"
    • Claimed demonstration of quantum coputational advantage using superconducting circuits and optical network
    • Comparison with classical algorithms is not simple\(\implies\)Latter part of this talk

LIGO

Reuters

Useful vs "Useless" quantum advantage

Outline

  • Classical foil

  • Provable quantum advantage

    • Correlation game (Bell's theorem)

    • Complexity theory primer

    • Quantum supremacy

  • Summary

Minimal introduction to my work

What is the classical theory we are comparing to?

Quantum Rules

Classical Rules

  • A quantum state is a vector \(|\psi\rangle\) in a Hilbert space
  • The state evolves according to the Schrödinger equation

 

 

  • The system can be found in state \(|e_j\rangle\) with probability \(|\langle e_j|\psi\rangle|^2\); after the measurement, the state is updated to \(|e_j\rangle\)
  • Independent systems combine via the tensor product
\displaystyle{i\hbar \frac{d|\psi\rangle}{dt} = H|\psi\rangle}
|\psi_1\rangle \otimes \cdots \otimes |\psi_n\rangle
  • A classical state is a point \((q,p)\) in a phase space
  • The state evolves according to Hamilton's equations

 

 

  • Measurements are trivial
  • Independent systems combine via the Cartesian product
\dot{q} = \displaystyle{\frac{\partial H}{\partial p}} \qquad \dot{p} = - \displaystyle{\frac{\partial H}{\partial q}}
(q_1, p_1, \dots, q_n,p_n)

Quantum Rules

  • A quantum state is a vector of complex numbers

 

 

 

  • The system can be found in state \(|e_j\rangle\) with probability \(|\alpha_j|^2\); after the measurement, the state is updated to \(|e_j\rangle\)
  • The evolution preserves the 2-norm
  • Independent systems combine via the tensor product

Probability Rules

  • A probability distribution is a vector of positive numbers 

 

 

 

  • The system can be found in state \(|e_j\rangle\) with probability \(p_j\); after the measurement, the state is updated to \(|e_j\rangle\)
  • The evolution preserves the 1-norm
  • Independent systems combine via the tensor product
|\psi\rangle = \begin{pmatrix} \alpha_1 \\ \vdots \\ \alpha_d \end{pmatrix}, \quad \sum_j |\alpha_j|^2 =1
|p\rangle = \begin{pmatrix} p_1 \\ \vdots \\ p_d \end{pmatrix}, \quad \sum_j p_j =1

2-norm

1-norm

Quantum theory is a generalization of probability theory using 2-norm!

Aaronson, arXiv:quant-ph/0401062

\begin{pmatrix} p \\ 1-p \end{pmatrix} \otimes \begin{pmatrix} q \\ 1-q \end{pmatrix} = \begin{pmatrix} pq \\ p(1-q) \\(1-p)q \\ (1-p)(1-q) \end{pmatrix}

Suppose I have two coins, one with probability \(p\) for head and another with probability \(q\) for head, their joint distribution indeed is the tensor product

HH
HT
TH
TT

Two Views of Quantum Theory

Quantum theory is a classical theory that restricts what you can know

Quantum

Classical?

Classical?

Quantum theory is an extension of classical theory

Hidden variable theory

The Correlation Game that Proves Einstein Wrong

◌ุอีกแล้ว

Q1

Q2

Q1

Q2

Correlation Game

Q1

Q2

Q1

Q2

Correlate

Anticorrelate

Correlation Game

P_{\mathrm{win, classical}} \le \frac{3}{4}
P_{\mathrm{win, quantum}} \lesssim 0.85

Correlate

Anticorrelate

Correlation Game

P_{\mathrm{win, classical}} \le \frac{3}{4}

First provable "useless" quantum advantage

P_{\mathrm{win, quantum}} \lesssim 0.85
\frac{|\uparrow_z\rangle\otimes |\downarrow_z\rangle - |\uparrow_z\rangle\otimes |\downarrow_z\rangle}{\sqrt{2}}

hackerdashery, Youtube

Complexity Theory Primer

Complexity Theory Primer

Seven bridges of Königsberg

traversing the graph without crossing the same edge twice

traversing the graph without visiting the same node twice

Graph problems:

Hamilton's icosian game

  • Generalize to \(\mathrm{Euler}(G)\) and \(\mathrm{Hamilton}(G)\) where a graph \(G\) with \(n\) edges defines an instance of the problem
  • If an Eulerian cycle exists, it can be found in \(\propto n\) steps. Meanwhile no efficient way to find a Hamiltonian path is known despite the similarity of the two problems!

Seven bridges of Königsberg

Complexity Theory Primer

Eulerian cycle

Hamiltonian cycle

Complexity Theory Primer

Computational complexity seeks to characterize the difficulty of mathematical problems based on the resource scaling of their algorithms with respect to the size of the problems

  • We say that a polynomial scaling \(\propto n^c\) is efficient
  • Usually the scaling given is an upper bound on runtime even in the worst instance
  • Eulerian cycle
  • Multiplication
  • Primality testing
  • Linear programming

Decision problem solvable in polynomial time

Decision problem verifiable in polynomial time

  • Hamiltonian cycle
  • Traveling salesman
  • Packing
  • Graph coloring

hackerdashery, Youtube

The Class NP

Conjecture: P\(\neq\)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

f(u) = \underbrace{(u_1 \vee u_2)} \wedge (u_2 \vee u_3) \wedge (u_3 \vee u_1)

Boolean formula (conjunctive normal form)

Clause

kSAT asks whether a given Boolean formula wherein each clause has \(k\) literals is satisfiable

  • kSAT is in NP (2SAT is in P, while 3SAT is NP-complete)
  • But note the asymmetry: there is no obvious short proof to convince you that a Boolean formula is unsatisfiable

Literals

\iff\exists u\, \mathrm{s.t.} f(u) = \mathrm{TRUE}
\iff \forall u\, \mathrm{s.t.} f(u) = \mathrm{FALSE}

Generalizing the Class NP

Problems with \(p\) alternative quantifiers (\(\exists,\forall\)) define the \(p\)-level of the polynomial hierarchy (PH), a generalization of NP

Conjecture: The PH does not collapse

Wikipedia

\Sigma_0 = \mathrm{P}
\Sigma_1 = \mathrm{NP}
\Sigma_p
\Sigma_{\infty}
\#\mathrm{P}

Beyond the top of this infinite tower lies the counting complexity class #P (Toda's theorem)

Counting is hard!

A #P problem asks for the number of solutions to an NP problem. Thus, it is at least as hard as the NP problem.

Generalizing the Class NP

Quantum Supremacy

Reuters

Southern China Morning Post

Quantum Supremacy

Q

p(x)

Quantum device

Classical simulator

C

Description of Q

Choice of a quantum program Q

q(x)\sim p(x)
\sum_{x\in\{0,1\}^n}|p(x)-q(x)|\le \epsilon

We want to rule out all such efficient classical simulator!

x\in\{0,1\}^n

output bitstring

Quantum Supremacy

C

Description of Q

q(x)\sim p(x)
\sum_{x\in\{0,1\}^n}|p(x)-q(x)|\le \epsilon

Recipe for proving quantum supremacy

  1. Choose a class of quantum programs
  2. Easy problem: show that approximating an ideal output probability \(p(x)\) is #P-hard
  3. Hard problem: connect 2. to the hardness of sampling from \(q(x)\)

Existence of an efficient classical simulator \(\implies\) Collapse of the PH

How Does This Work?

Non-collapse of the PH \(\implies\) no such simulator exist

End goal:

Let us ignore sampling for a moment, and consider the complexity of calculating, say \(q(x_1=1)\), the probability that the first bit of outcome is 1

Counting the number of random bitstring that give \(f(r)=1\) is a #P-hard problem!

Hidden bitstring

q(x_1=1) = \displaystyle{\frac{1}{2^m}\sum_{r\in\{0,1\}^m} f(r)}

Classical randomness can always be modeled as averaging over random hidden bitstrings

C

q(x)
\Sigma_0 = \mathrm{P}
\Sigma_1 = \mathrm{NP}
\Sigma_p

How Does This Work?

  • Exactly calculating an output probability of a classical machine is #P-hard
  • But approximating such a probability is much easier \(\impliedby\) Stockmeyer's approximate counting
\Sigma_{\infty}
\#\mathrm{P}
\mathrm{approx\#P}
\exists H \, \forall z\neq z' \in \mathrm{Acc(r) s.t.} H(z) \neq H(z')

Stockmeyer's approximate counting ultimately decides a predicate such as 

which contains a few logical quantifiers, and thus lies in a low level of the PH

Approximate counting is "easy" if you can grab the hidden variables

Recipe for proving quantum supremacy

  1. Choose a class of quantum programs
  2. Easy problem: show that approximating an ideal output probability \(p(x)\) is #P-hard
  3. Hard problem: Show that approximating \(q(x)\) lets you approximate \(p(x)\) as well

How Does This Work?

C

Description of Q

q(x)\sim p(x)
\sum_{x\in\{0,1\}^n}|p(x)-q(x)|\le \epsilon

Recipe for proving quantum supremacy

  1. Choose a class of quantum programs
  2. Easy problem: show that approximating an ideal output probability \(p(x)\) is #P-hard
  3. Hard problem: Show that approximating \(q(x)\) lets you approximate \(p(x)\) as well

How Does This Work?

\Sigma_0 = \mathrm{P}
\Sigma_1 = \mathrm{NP}
\Sigma_{\infty}
\#\mathrm{P}
\mathrm{approx\#P}

Solve

Collapse of the PH

Even approximating an output probability of a quantum machine is #P-hard; there is no hidden variable to grab!

Take-Home Messages

  • Provable quantum advantage tasks are designed to be provable with available mathematical techniques, although they maybe "useless"
  • Provable quantum advantage is about proving the limitation of classical information processing
  • Computation complexity plays a vital part in the current proof of quantum supremacy

Hartnoll, Sachdev, Takayanagi, et al., Quantum connections, Nat Rev Phys 2021

  • Simulation of strongly correlated quantum systems/field theories
  • Entanglement entropy as an organizing principle in high-energy and condensed matter physics
  • SYK model and black holes, holographic duality, emergent gravity,  exotic matters from quantum error correcting codes

Quantum Information Science

Two-way flow of ideas between physics and computer/information science

QIS\(\implies\)Physics

 Physics \(+\) Computer science

  • Completely new kind of information processing
  • Computational complexity dependent on physical law
  • Quantum theory from information-theoretic principles

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.

P.A.M. Dirac 1929

Michael R. Fellows*, 1991 

Computer science is not about machines, in the same way that astronomy is not about telescopes.

*often misattributed to Edsger Dijkstra