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
- 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
- Choose a class of quantum programs
- Easy problem: show that approximating an ideal output probability \(p(x)\) is #P-hard
- 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
- Choose a class of quantum programs
- Easy problem: show that approximating an ideal output probability \(p(x)\) is #P-hard
- 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
- Choose a class of quantum programs
- Easy problem: show that approximating an ideal output probability \(p(x)\) is #P-hard
- 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
Quantum Advantage Mahasarakam
By Ninnat Dangniam
