Prof. Riku JänttiDepartment of Communications and Networking

Qunatum MachineLearning

Contents

1. Motivation2. Very brief introduction to quantum theory3. Quantum computing4. Quantum machine learning

Simple quantum classifier5. Application to Quantum Backscatter Communications6. Conclusions

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1. Motivation

Machine learning• “Machine learning (ML) is the scientific

study of algorithms and statisticalmodels that computer systems use toperform a specific task without usingexplicit instructions, relying on patternsand inference instead.” Wikipedia.

• One typical machine learning problem isclassification of high dimensional data.

• In supervised learning, we have a set oftraining data with correct labels,

• In unsupervised learning, the algorithmbuilds a mathematical model from a set ofdata which contains only inputs and nodesired output labels.

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Support vector machine

K-means clustering

Machine learning• Machine learning has become very popular and found

applications almost all areas of our lives.• As the amount of data grows so do the number of required

computations and consumed power

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Strubell, Emma, Ananya Ganesh, and Andrew McCallum. "Energy and Policy Considerations for DeepLearning in NLP." arXiv preprint arXiv:1906.02243 (2019).

Energy and time cost to train certain natural language processing models

Quantum machine learning

• Quantum machine learning make use ofquantum computing to speed up certain type ofalgorithms.

• Quantum computers on the other handconsume much more power than traditionalcomputers (due to cooling to mKtemperatures). Hence, energy saving wouldmanifest itself only for truly big data problems.

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Biamonte, Jacob, et al. "Quantum machinelearning." Nature 549.7671 (2017): 195-202.

Quantum machine learning

• Quantum sensing and metrology develops new high-precisionmeasurement systems.

• These systems generate quantum data that could be directlyprocessed by quantum computers without having to convert it toclassical data (by measurement).

• Quantum computers can then run quantum machine learningalgorithms to process the measurement data in a efficientmanner.

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2. A very briefintroduction toquantumcomputing

NotationVector notationxÎ d Row vector in ddimensional Hilbert space H

yH Conjugate transpose of y

yHx Inner product

Ax linear mappingA=[aij] n by n matrixaij= bi

HAbj

A has a basis {bj}Ax=lx l eigenvalue of ATr{A}=Siaii Matrix trace

Dirac’s notation⟩| ket

⟨ | bra

braket

⟩| linear operatoris anoperator

aij= ⟨ | ⟩|has a basis {| }⟩| =l ⟩| l eigenvalue of

Tr = = ∑ ⟨ | ⟩|Operator trace

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Heisenberg and Schrödinger pictures

In physics, the Heisenberg picture is a formulation (largely due toWerner Heisenberg in 1925) of quantum mechanics in which theoperators (observables and others) incorporate a dependency ontime, but the state vectors are time-independent, an arbitrary fixedbasis rigidly underlying the theory.

In physics, the Schrödinger picture is a formulation of quantummechanics in which the state vectors evolve in time, but theoperators (observables and others) are constant with respect totime.

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Quantum mechanics

An isolated system can be completely specifies by a state, given bya unit vector ⟩|y in Hilbert space H.

.

If ⟩|y1 , ⟩|y2 , … , ⟩|y are possible states of the system, the ⟩|y canbe expressed as a state superposition

Î

∑ =1

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y y =1

⟩|y =∑ ⟩|y ,

Qubit and Qudit in Heisenber picture

Qubit is a coherent superposition of two orthonormal basis states|0> and |1>:

Qudit is a coherent superposition of more than two orthonomalstates

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⟩|y = ⟩|0 + ⟩|1

=2

= f2

⟩|y =∑ ⟩| ,

Measureing a qubit

A Qubit: ⟩|y = ⟩|0 + |1 and observables

Measurement in {|0>, |1>} basis

0 = ⟩|0 ⟨ |0 => ⟨ |y 0 ⟩|y =

1 = ⟩|1 ⟨ |1 => ⟨y| 1 ⟩|y =

• Measurement in Bell basis ⟩|0 ± ⟩|1

+= ⟩|0 + ⟩|1 ⟨ |0 +⟨ |1 => ⟨y|

+⟩|y = 1+cos

−= ⟩|0 − ⟩|1 ⟨ |0 −⟨ |1 => ⟨y|

−⟩|y = 1−cos

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Independent of the phase

Measures thephase

Quantum mechanics

Let ⟩|y be a state of a quantum system A and ⟩|y be a state ofquantum system B. If the two systems are independent, then thejoint state is given by ⟩|y = ⟩|y Ä ⟩|y = ⟩|y ⟩|y where Ä denotesthe Kroneker product.

Example two independent qubits

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⟩|y = ⟩|y Ä ⟩|y= ⟩|0 + ⟩|0 Ä ⟩|0 + ⟩|0 =

= ⟩|0 ⟩|0 B+ ⟩|0 ⟩|1 + ⟩|1 ⟩|1 + ⟩|1 ⟩|1

Quantum gates

We can perform unitary operation on the quantum states withouthaving to measure them. Such operations are reversible.

† ⟩|y = ⟩|yExample Unitary gate acting on single qubit

⟩|0 =12 00 ⟩|0 + ⟩10|1

⟩|0 =12 0 ⟩|0 + ⟩11|1

= 00 0

10 11is Unitary matrix UHU=I

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=

3.QuantumComputing

Programming a quantum computer• Programming a quantum computer = defining quantum

gates that perform the desired operations.• Similar to early days of digital computing using digital

circuits.

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Single qubit quantum gates

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⟩|0 =12

⟩|0 + ⟩|1

⟩|1 =12

⟩|0 − ⟩|1

Hadamard

X

Y

Pauli-X / NOT gate

Pauli-Y

s ⟩|0 = ⟩|1 s ⟩|1 = ⟩|0

s ⟩|0 = ⟩|1 s ⟩|1 = − ⟩|0

ZPauli-Z / Rp phase shift gate

Rp

s ⟩|0 = ⟩|0 s ⟩|1 = − ⟩|1

Squareroot NOT⟩|0 =

12 (1 + ) ⟩|0 +(1−i) ⟩|1

⟩1 =12 (1− ) ⟩|0 +(1+i) ⟩|1

= =

Phase shifter Rf

Rf ⟩|0 = ⟩|0Rf ⟩|1 = f ⟩|1

Rf

Identity s ⟩|0 = ⟩|0 s ⟩|1 = ⟩|1

Two qubit (qudit) gates

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SWAP gate ⟩|0 ⟩|0 = ⟩|0 ⟩|0⟩|0 ⟩|1 = ⟩|1 ⟩|0⟩|1 ⟩|0 = ⟩|0 ⟩|1⟩|1 ⟩|1 = ⟩|1 ⟩|1

Controlled NOT⟩|0 ⟩|0 = ⟩|0 ⟩|0⟩|0 ⟩|1 = ⟩|0 ⟩|1⟩|1 ⟩|0 = ⟩|1 ⟩|1⟩|1 ⟩|1 = ⟩|1 ⟩|0

Other two qubit gates

Square root swap

Controlled unitary

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Decomposition to elementary gates

⟩|0 ⟩|0 = ⟩|0 ⟩|0⟩|0 ⟩|1 = ⟩|0 ⟩|1⟩|1 ⟩|0 =| ⟩1 |0 =| ⟩1 ⟩( 00|0 + ⟩10|1 )⟩|1 ⟩|1 = ⟩|1 |1 =| ⟩1 ⟩( 01|0 + ⟩11|1 )

Multi-qubit gates

… can be decomposed to single and two qubit gates.

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Fredkin gate / Controlled SWAP

⟩|0 ⟩| ⟩| = ⟩|0 ⟩| ⟩|⟩|0 ⟩| ⟩| = ⟩|0 ⟩| ⟩|

4.QuantumMachine learning

Quantum machine learningapproaches• Quantum computing techniques that can be utilized for ML

• Linear algebra simulation with quantum amplitudes• Grover search• Quantum-enhanced reinforcement learning• Quantum annealing• Quantum neural networks• Hidden Quantum Markov Models• Fully quantum machine learning• …

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Linear algebra simulation withquantum amplitudesLarge linear algebraic functions canbe performed efficiently using aquantum computer.

Ax=b => x=A-1b

With quantum computer is not easyto solve for x directly but rather solvefor Hermitian quadratic forms xHMx.

Many machine learning schemesrequire linear algebraic operations tobe solved in large dimensionalspace. With quantum computer, wecan obtain speedup.

For a sparse matrix A having a lowcondition number k, classicalalgorithm has runtime O(Nk) and thequantum algorithm O(log(N)k^2)

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Cai XD, Weedbrook C, Su ZE, Chen MC, Gu M, Zhu MJ, Li L, Liu NL, Lu CY,Pan JW. Experimental quantum computing to solve systems of linearequations. Physical review letters. 2013 Jun 6;110(23):230501.

The simplest quantum linear operator:Correlator• Inner product aTb between two vectors a and b is used in linear

classifier as a measure of similarity. The two vectors aredissimilar if their inner product is small.

• In Quantum case, we can use ⟩⟨ | as a similarity measurebetween two quantum states ⟩| and ⟩| .

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b

a

aTb

Swap test algorithm

Quantum circuit realizing the swap test

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⟩|0 ⟩|y ab+ ⟩|1 ⟩|y ab

⟩|y ab

⟩|0 ⟩|y ab+ ⟩|1 ⟩|y ab

⟩|1 (1) ⟩|y ab+ (2) ⟩|0 ⟩|y ab

j Pr{0}=⟨ |y ( ) ⟩|y= 1 + |

Application to sending classicalinformation over a quantum channelTransmitter encodes classical information to a bell state:⟩| = ⟩|0 + ⟩|1 x=-1,1

The channel induces unknown phase shift . The received qubit isthus

⟩| ; = ⟩|0 +x |1

To probe the channel we send a pilot qubit

⟩|1 = ⟩|0 + ⟩|1

Correlator receiver: SWAP test for ⟩| ; ⟩| ; 1

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Application to sending classicalinformation over a quantum channelMeasurement Pr{0} = 1 + ; | ; 1

; | ; 1 = 1 + = 1 = 10 = −1

Protocol:Repeat the transmission M times and perform hypothesis test overbinomial random variable

H0: p=0.5 (-1 transitted)H1: p¹0.5 (+1 transmited)

Repeating the protocol 6 times would give us 5% error probability.

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Quantum neural networks

• Basic building block of a neural network is a perceptron that consists of linearoperation followed by nonlinear mapping.

• In quantum computing, we only have unitary operators available so we need to usemeasurement to realize the nonlinearity.

• A central goal of quantum neural network research is to improve the computing timeof the training phase of artificial neural networks through a clever exploitation ofquantum effects.

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Application toquanutumcommunciations

Quantum backscatter communications

• Quantum illumination is utilizedis utilized in backscattercommunications to obtain 6 dBgain in the bit error exponent.

• The target is antenna withknown properties.

• Higher thermal noise isexpected than in the radarcase (antennas pointedtowards the warm groundinstead of cool sky).

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R. DiCandia, R. Jäntti, R. Duan, J. Lietzen, H. Kalifa, and K. Ruttik, ”Quantum BackscatterCommunications: A New Paradigm,” in Proc. ISWCS 2018

K. Hany and R. Jäntti, “Quantum backscatter communication with photon number states,”Workshop on Quantum Communications and Information Technology (QCIT'18) at IEEEGlobecom 2018, December 9-13, Abu Dhabi, 2018.

SI

Entangledphoton pairgeneration

Txantenna

Rxantenna

Receiver

Backscatterantenna

Applications of ML?

• In certain cases, we can obtain the received signal directlyas qubits which could be processed using quantumcomputer.

• This allows us to implement receiver signal processingalgorithms using e.g. quantum machine learning techniques.

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Conclusions

Conclusions

• Quantum machine learning is rapidly developing branch ofquantum computation.

• It can be utilized to speed up classical machine learningtasks as well as perform fully quantum machine learningtasks.

• The speedup gains are likely to be significant for big data…but would require large quantum computers which arenot yet available.

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Department of Communicationsand Networking (Comnet)