Quantum computing

(in the cloud)

Jonas S. Neergaard-Nielsen

QuantumDTU Summer School on Quantum Technology

15 August 2019

Slides (with some interactivity) available at

bit.ly/qt-school19

On the menu

  • brief overview
  • the ingredients: qubits, gates, measurements
  • the dishes: circuits, algorithms
  • the bad aftertaste: errors
  • the other restaurants in town: NISQ, quantum players, open tools

The poster child

of quantum technology

largest hype

largest investments

largest potential value

longest time horizon

Computational complexity

Some computational problems are hard

- meaning, very VERY hard!

You do not mess with exponential scaling...

Because of superposition and entanglement, quantum systems scale exponentially in qubit size

Computational complexity

I love when someone can make a 10 min video that conceptually explains a CS topic better than my grossly overpaid professor that can't take the time to make decent slides.

YouTube comment

Simulating quantum

Nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy.

Quantum supremacy

Classical systems cannot in general simulate quantum systems efficiently.

 

We cannot yet prove this claim, either mathematically or experimentally, but we have reason to believe it is true; arguably, it is one of the most interesting distinctions ever made between quantum and classical.

It means that well controlled large quantum systems may “surpass understanding,” behaving in ways we find surprising and delightful.

 

We therefore hope to hasten the onset of the era of quantum supremacy, when we will be able to perform tasks with controlled quantum systems going beyond what can be achieved with ordinary digital computers.

Low-level building blocks

  • initialized qubits
  • 1-qubit gates
  • 2- and/or 3-qubit gates
  • measurements (+ possibly feed-forward)

Quantum circuits

Qubits

|\psi\rangle=\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle
|\psi\rangle
\tfrac{1}{\sqrt{2}}(|0\rangle+|1\rangle)
\tfrac{1}{\sqrt{2}}(|0\rangle-|1\rangle)
\tfrac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)
\tfrac{1}{\sqrt{2}}(|0\rangle-|1\rangle)

Gates

\left(\!\! \begin{array}{cc} 1 & 0 \\ 0 & e^{i\pi/4} \end{array}\!\! \right)
\left(\!\! \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\!\! \right)
\left(\!\! \begin{array}{cc} 0 & -i \\ i & 0 \end{array}\!\! \right)
\left(\!\! \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\!\! \right)
\left(\!\! \begin{array}{cc} 1 & 0 \\ 0 & i \end{array}\!\! \right)

X

Y

Z

S

T

H

\begin{aligned} \hat{X}|0\rangle &= |1\rangle \\ \hat{X}|1\rangle &= |0\rangle \end{aligned}
\begin{aligned} \hat{Y}|0\rangle &= i|1\rangle \\ \hat{Y}|1\rangle &= -i|0\rangle \end{aligned}
\begin{aligned} \hat{Z}|0\rangle &= |0\rangle \\ \hat{Z}|1\rangle &= -|1\rangle \end{aligned}
\begin{aligned} \hat{S}|0\rangle &= |0\rangle \\ \hat{S}|1\rangle &= i|1\rangle \end{aligned}
\begin{aligned} \hat{T}|0\rangle &= |0\rangle \\ \hat{T}|1\rangle &= \tfrac{1+i}{\sqrt{2}}|1\rangle \end{aligned}
\begin{aligned} \hat{H}|0\rangle &= \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \\ \hat{H}|1\rangle &= \tfrac{1}{\sqrt{2}}(|0\rangle - |1\rangle) \end{aligned}

X (or NOT) gate - flips the basis states

Y gate - flips around the Y-axis

Z gate - π phase shift (flip around Z)

phase gate - π/2 phase shift

π/4 gate

Hadamard gate - changes basis

|\psi\rangle

$$ \frac{1}{\sqrt{2}}\!\left(\!\!\begin{array}{cc}1 & 1 \\1 & -1\end{array}\!\!\right) $$

Gates

\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right)
|\psi\rangle
|\psi\rangle
|\psi\rangle

U

SWAP gate - does what it says

CNOT - controlled-NOT (C-X)

C-U - controlled unitary

Toffoli - controlled-controlled-NOT

\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)
\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & u_{00} & u_{01} \\ 0 & 0 & u_{10} & u_{11} \end{array} \right)

vectorcise!

Measurements

|\psi\rangle
|0\rangle
|0\rangle

H

H

Z

+ feed-forward

Quantum teleportation

source: Quirk

Quantum Fourier transform 

(core part of Shor's factoring algorithm)

$$ \hat{U}_{QFT} |x\rangle = \frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^n-1} \exp\! \left( i \frac{2\pi x k}{2^n} \right) |k\rangle $$

Algorithms

Deutsch-Jozsa

  • 1992 - first algorithm to show speedup over classical
    - at least over deterministic approach (P, not BPP)
  • illustrates some central ideas in Q.C.
    • inputs in superposition
    • ancilla qubits
    • oracle queries
    • phase kick-back
    • answer by interference
  • utterly useless!

Deutsch-Jozsa

task:  given an unknown function \( f: \{0,1\}^n \rightarrow \{0,1\} \) which is promised to be either constant or balanced (half of function values are 0, the other half 1),

determine with certainty which it is - constant or balanced

"oracle" - implements \( \hat{U}_f |x\rangle |y\rangle = |x\rangle |y \oplus f(x) \rangle \)

input \( |x\rangle \), encoding the number \( x \in [0; {2^n-1}] \)

ancilla qubit \( |y\rangle \)

Deutsch-Jozsa

whiteboard

&

quantum computer - in the cloud!

IBM Q System One

Quantum algorithm zoo

Factoring Superpolynomial
Discrete-log Superpolynomial
Pell's Equation Superpolynomial
Principal Ideal Superpolynomial
Unit Group Superpolynomial
Class Group Superpolynomial
Gauss Sums Superpolynomial
Primality Proving Polynomial
Solving Exponential Congruences Polynomial
Verifying Matrix Products Polynomial
Subset-sum Polynomial
Decoding Varies
Constraint Satisfaction Polynomial
Quantum Cryptanalysis Various
Searching Polynomial
Abelian Hidden Subgroup Superpolynomial
Non-Abelian Hidden Subgroup Superpolynomial
Deutsch-Jozsa Exponential over P, none over BPP
Formula Evaluation Polynomial
Hidden Shift Superpolynomial

Bernstein-Vazirani Polynomial Directly, Superpolynomial Recursively
Matrix Elements of Group Representations Superpolynomial
Gradients, Structured Search, and Learning Polynomials Polynomial
Polynomial interpolation Varies
Pattern matching Superpolynomial
Linear Systems Superpolynomial
Ordered Search Constant factor
Graph Properties in Adjacency Matrix Model Polynomial
Graph Properties in Adjacency List Model Polynomial
Welded Tree Superpolynomial
Collision Finding and Element Distinctness Polynomial
Graph Collision Polynomial
Matrix Commutativity Polynomial
Group Commutativity Polynomial
Hidden Nonlinear Structures Superpolynomial
Center of Radial Function Polynomial
Group Order and Membership Superpolynomial
Group Isomorphism Superpolynomial
Matrix Multiplication over Semirings Polynomial
Quantum Approximate Optimization Superpolynomial

Statistical Difference Polynomial
Finite Rings and Ideals Superpolynomial
Counterfeit Coins Polynomial
Matrix Rank Polynomial
Subset finding Polynomial
Search with Wildcards Polynomial
Network flows Polynomial
Electrical Resistance Exponential
Machine Learning Varies
Junta Testing and Group Testing Polynomial
Quantum Simulation Superpolynomial
Knot Invariants Superpolynomial
Three-manifold Invariants Superpolynomial
Partition Functions Superpolynomial
Adiabatic Algorithms Unknown
Semidefinite Programming Superpolynomial
Zeta Functions Superpolynomial
Weight Enumerators Superpolynomial
Simulated Annealing Polynomial
String Rewriting Superpolynomial
Matrix Powers Superpolynomial

Quantum error correction

Logical vs. physical qubits

Encode logical qubits in multiple physical qubits

- local errors can then be detected and corrected

these states are +1 eigenstates of the stabilizer operators \( ZZI = \hat{Z} \otimes \hat{Z} \otimes \hat{I} \) and \( IZZ = \hat{I} \otimes \hat{Z} \otimes \hat{Z} \)

If one of the 3 qubits has flipped, i.e. \( XII|\tilde\psi\rangle \), \( IXI|\tilde\psi\rangle \) or \( IIX|\tilde\psi\rangle \),
measurement of \(ZZI\) and \(IZZ\) - the error syndrome - will reveal which qubit flipped

|\psi\rangle=\alpha |0\rangle + \beta |1\rangle \quad \rightarrow \quad |\tilde\psi\rangle = \alpha |000\rangle + \beta |111\rangle

Exercise 1: Calculate expectation values \(\langle ZZI \rangle \) and \(\langle IZZ \rangle \) for all 3 bit flips.

Exercise 2: Design a circuit that encodes the logical qubit and performs the syndrome measurement using 2 ancilla qubits

Fault-tolerant QC

Schemes with 5 or more qubits can correct arbitrary single-qubit errors

More advanced schemes with better properties exist, e.g.

  • surface codes - high error thresholds around ~1%
  • bosonic codes - encoding in high-dimensional harmonic oscillators

With error correction and error rates below a certain threshold, fault-tolerant quantum computing is possible, in principle

Still, ~1000 physical qubits required for one good, protected logical qubit

- many millions of physical qubits required for any reasonably useful use case

Current quantum computing landscape

Even with fault-tolerant quantum computing still a rather distant dream, we are now
entering a pivotal new era in quantum technology. [...]

I needed a name to describe
this impending new era, so I made up a word: NISQ. This stands for Noisy Intermediate-
Scale Quantum.

John Preskill

Quantum 2, 79 (2018)

NISQ

NISQ

John Martinis, Google - via QWA

NISQ

  • quantum simulation
    • chemistry
    • annealing, boson sampling
  • optimization problems
    • hybrid quantum-classical (e.g. VQE)
    • quantum semi-definite programming
  • machine learning

- and simply learning about quantum computers!

Rigetti - via QWA

Turing

Universal Quantum

Xanadu

Xofia

Zapata Computing

ZY4

 

Accenture

Airbus Group

Alibaba Group

Archer Exploration Limited

AT&T (INQNET Alliance)

Atos Quantum

Baidu

Booz Allen Hamilton

Google Quantum AI Lab

Honeywell

HP

IBM

Intel

JSR Corporation

Lockheed Martin

Microsoft

Mitsubishi Electric

Nokia Bell Labs

Northrup Grumman

NTT Basic Research Laboratories

Quantum Computing Inc.

Raytheon

Toshiba – Cambridge Research Laboratory

1QBit

A*Quantum

Agnostiq

Aliro Technologies

Alpine Quantum Technologies, GmbH

Ankh.1

Anyon Systems Inc.

ApexQubit

AppliedQubit

Artiste-qb.net

Atom Computing

Aurora Quantum Technologies

Automatski

BardeenQ Labs

Beit.tech

Bleximo

Boxcat

Black Brane Systems

bra-ket science

BraneCell

Cambridge Quantum Computing Ltd

Chicago Quantum

ColdQuanta

CryptoNext Security

D Slit Technologies

D-Wave Systems Inc.

Delft Circuits

EeroQ

Elyah

Entanglement Partners

Entropica Labs

evolutionQ

GTN LTD

H-Bar Consultants

Horizon Quantum Computing

HQS Quantum Simulations

Huawei

ID Quantique

InfiniQuant

IonQ

IQM Finland

ISARA

JoS QUANTUM

Ketita Labs

KETS Quantum Security

kiutra

Labber Quantum

M-Labs

Magiq

MDR

Multiverse Computing

NetraMark

NextGenQ

Nordic Quantum Computing Group

Nu Quantum

Origin Quantum Computing

Oxford Quantum Circuits

Pasqal

Phase Space Computing

PhaseCraft

Post-Quantum (PQ Solutions)

ProteinQure

PsiQuantum

Q&I

Q-Ctrl

Q-Lion

Qasky

QbitLogic

Qblox

QC Ware

QEYnet

Qilimanjaro Quantum Hub

Qindom

Qnami

Qrithm

Qrypt

Qu & Co

Quandela

Quantastica

QuantFi

QuantiCor Security

Quantika

Quantopo

Quantum Benchmark Inc.

Quantum Circuits, Inc.

Quantum Factory

Quantum Impenetrable

Quantum Machines

Quantum Motion Technologies

Quantum Phi

Quantum Thought

Quantum Xchange

QuantumCTek

QuantumX

Quantumz.io

QuBalt GmbH

Qubitekk

Qubitera, LLC

Qubit Reset LLC

QuDot

Quintessence Labs

QuiX

Qulab

QunaSys

Qunnect

QuNu Labs

QURECA

Quside

QxBranch

Rahko

Rigetti Computing

Riverlane

SeeQC.EU

SHYN

Sigma-i

Silicon Quantum Computing

softwareQ

Solid State AI

Sparrow Quantum

SpeQtral

Spin Quantum Tech

Strangeworks

Tokyo Quantum Computing

TundraSystems Global LTD

Be aware of the hype

But - they're already very useful!

ProjectQ is an open-source software framework for quantum computing started at ETH Zurich. It allows users to implement their quantum programs in Python using a powerful and intuitive syntax. ProjectQ can then translate these programs to any type of back-end, be it a simulator run on a classical computer or an actual quantum chip.

many other available tools...

It's your playground!