An Introduction to Quantum Computing and Quantum Random Number Generators

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An introduction to Quantum Computing and Quantum Random Number Generators

Giuseppe Vallone

email: [email protected]

INFN and The Future of Scientiﬁc Computing - 4 May 2018 Summary

1 Introduction

2 Quantum Computing Circuit model Alternatives to circuit model Implementations

3 Quantum Random Number Generators

4 Conclusions

Pag. 2 Intro Quantum Computing QRNG Conclusions Summary

1 Introduction

2 Quantum Computing Circuit model Alternatives to circuit model Implementations

3 Quantum Random Number Generators

4 Conclusions

Pag. 3 Intro Quantum Computing QRNG Conclusions What is Quantum Information?

Information Theory Quantum Mechanics

⇔

Merging two big XXth century revolutions: information theory (Shannon, Turing) and Quantum Mechanics.

Pag. 4 Intro Quantum Computing QRNG Conclusions Examples of applications

Quantum computer Quantum cryptography

Quantum metrology

Pag. 5 Intro Quantum Computing QRNG Conclusions Examples of applications Quantum sensing Quantum imaging

Quantum random number Quantum simulation generation

Pag. 6 Intro Quantum Computing QRNG Conclusions But...

...be aware of fake!

Pag. 7 Intro Quantum Computing QRNG Conclusions Summary

1 Introduction

2 Quantum Computing Circuit model Alternatives to circuit model Implementations

3 Quantum Random Number Generators

4 Conclusions

Pag. 8 Intro Quantum Computing QRNG Conclusions Summary

1 Introduction

2 Quantum Computing Circuit model Alternatives to circuit model Implementations

3 Quantum Random Number Generators

4 Conclusions

Pag. 9 Intro Quantum Computing QRNG Conclusions One-slide review of classical computation

Input Output n bits m bits

Universal gates: any function

f : {0, 1}n → {0, 1}m

can be built from two elementary gates: NAND and COPY.

Pag. 10 I Superposition principle: if |ψ1i and |ψ2i are physical states, any linear combination is a physical state:

|Ψi = a|ψ1i + b|ψ2i a, b ∈ C

I From classical bit (two orthogonal states |0i and |1i) to quantum-bit , or qubit:

2 2 |ψi = α|0i + β|1i α, β ∈ C , |α| + |β| = 1

Intro Quantum Computing QRNG Conclusions Quantum computer: superposition principle

I Quantum mechanics: physical states are represented as vectors |ψi

Pag. 11 I From classical bit (two orthogonal states |0i and |1i) to quantum-bit , or qubit:

2 2 |ψi = α|0i + β|1i α, β ∈ C , |α| + |β| = 1

Intro Quantum Computing QRNG Conclusions Quantum computer: superposition principle

I Quantum mechanics: physical states are represented as vectors |ψi

I Superposition principle: if |ψ1i and |ψ2i are physical states, any linear combination is a physical state:

|Ψi = a|ψ1i + b|ψ2i a, b ∈ C

Pag. 11 Intro Quantum Computing QRNG Conclusions Quantum computer: superposition principle

I Quantum mechanics: physical states are represented as vectors |ψi

I Superposition principle: if |ψ1i and |ψ2i are physical states, any linear combination is a physical state:

|Ψi = a|ψ1i + b|ψ2i a, b ∈ C

I From classical bit (two orthogonal states |0i and |1i) to quantum-bit , or qubit:

2 2 |ψi = α|0i + β|1i α, β ∈ C , |α| + |β| = 1

Pag. 11 Intro Quantum Computing QRNG Conclusions State superposition

above 10000 AMU, 810 atoms Matter–wave interference of particles selected from a molecular library with masses exceeding 10000 amu, Phys. Chem. Chem. Phys. 15, 14696 (2013)

Pag. 12 I Manipulate the state into |Ψouti = U|Ψini. U is the algorithm

I Measure each qubit (usually in the computational basis)

Intro Quantum Computing QRNG Conclusions Circuit model of quantum computation

0 1 Input 0 1 Output n n bits qubits 0 1

I Prepare the system into |Ψini = |ψ0i ⊗ |ψ0i ⊗ · · · ⊗ |ψn−1i

Pag. 13 I Measure each qubit (usually in the computational basis)

Intro Quantum Computing QRNG Conclusions Circuit model of quantum computation

0 1 Input 0 1 Output n n bits qubits 0 1

I Prepare the system into |Ψini = |ψ0i ⊗ |ψ0i ⊗ · · · ⊗ |ψn−1i

I Manipulate the state into |Ψouti = U|Ψini. U is the algorithm

Pag. 13 Intro Quantum Computing QRNG Conclusions Circuit model of quantum computation

0 1 Input 0 1 Output n n bits qubits 0 1

I Prepare the system into |Ψini = |ψ0i ⊗ |ψ0i ⊗ · · · ⊗ |ψn−1i

I Manipulate the state into |Ψouti = U|Ψini. U is the algorithm

I Measure each qubit (usually in the computational basis)

Pag. 13 Intro Quantum Computing QRNG Conclusions Universal gates

Any quantum algorithm U can be built by the following universal gates:

Hadamard Single qubit rotation CNOT gate

ϕ 1 1 i σz   H = √1 Rz(ϕ) = e 2 1 0 0 0 2 1 −1 i ϕ ! 0 1 0 0 e 2 0   = −i ϕ 0 0 0 1 0 e 2 0 0 1 0

Pag. 14 Intro Quantum Computing QRNG Conclusions Shor algorithm

Problem 1: factoring Given an integer number N, ﬁnd its prime factors. n1/3(log n)2/3 The best classical algorithm requires O(e 2 ) with n = log2N

Problem 2: order ﬁnding Choose a generic a such that 1 < a < N. The order r of a modulo N is the minimum positive integer r such that ar = 1 mod N. Hard (probably superpolinomial) problem.

Equivalence By using O(n3) classical operations Problem 1 can be reformulated as Problem 2

Pag. 15 Intro Quantum Computing QRNG Conclusions Shor algorithm

Order-ﬁnding by quantum computer

x Va : |xi|yi → |xi|y ⊕ a mod Ni

3 I The Va operation requires O(n ) gates 2 I DFT requires O(n ) gates

Pag. 16 Intro Quantum Computing QRNG Conclusions Grover algorithm

problem: ﬁnd x0 given a quantum black box Uω such that ( −|xi for x = x0 Uω|xi = |xi for x 6= x0

√ x0 is found with high probability using O( N) evaluations A classical computer cannot solve the problem with less than O(N) operations

Pag. 17 Intro Quantum Computing QRNG Conclusions Summary

1 Introduction

2 Quantum Computing Circuit model Alternatives to circuit model Implementations

3 Quantum Random Number Generators

4 Conclusions

Pag. 18 Intro Quantum Computing QRNG Conclusions One-way quantum computer

G. Vallone, et al., One-way quantum computation with two-photon multiqubit cluster states, Phys. Rev. A 78, 042335 (2008)

Pag. 19 Intro Quantum Computing QRNG Conclusions Quantum simulation

Quantum Simulation, Rev. Mod. Phys. 86, 154 (2014)

Pag. 20 Intro Quantum Computing QRNG Conclusions Adiabatic Quantum Computing

H(t) = (1 − t)H0 + tHP 1 T = O( 2 ) gmin

Adiabatic Quantum Computing, Rev. Mod. Phys. 90, 015002 (2018)

Pag. 21 Intro Quantum Computing QRNG Conclusions Summary

1 Introduction

2 Quantum Computing Circuit model Alternatives to circuit model Implementations

3 Quantum Random Number Generators

4 Conclusions

Pag. 22 Intro Quantum Computing QRNG Conclusions Photons

Reprogrammable optical circuit that implements all possible linear optical protocols up to the size of the circuit (six photon input)

Universal linear optics, Science 349, 711 (2017)

Pag. 23 Intro Quantum Computing QRNG Conclusions Trapped ion

Quantum computers,Nature 464, 45 (2010)

Pag. 24 Intro Quantum Computing QRNG Conclusions Ultracol atoms in lattice

Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017)

Pag. 25 Intro Quantum Computing QRNG Conclusions Nuclear Magnetic Resonance (NMR)

NMR techniques for quantum control and computation, Rev. Mod. Phys. 76, 1037 (2005)

Pag. 26 Intro Quantum Computing QRNG Conclusions Quantum dots

Quantum computers, Nature 464, 45 (2010)

Pag. 27 Intro Quantum Computing QRNG Conclusions Superconducting qubits

Quantum computers, Nature 464, 45 (2010)

Pag. 28 Intro Quantum Computing QRNG Conclusions D-wave

Quantum annealing: ﬁnding the global minimum of a given function by using quantum ﬂuctuations.

Pag. 29 Intro Quantum Computing QRNG Conclusions Google

Superconducting qubit processor

A blueprint for demonstrating quantum supremacy with superconducting qubits, [arXiv: 1709.06678] Pag. 30 Intro Quantum Computing QRNG Conclusions IBM

IBM’s quantum processors made up of superconducting transmon qubits

https://quantumexperience.ng.bluemix.net/qx/qasm

Pag. 31 Intro Quantum Computing QRNG Conclusions Microsoft

“Our approach focuses on topological quantum computing through Majorana fermions, which promise to yield fast, stable quantum bits, also known as qubits.”

https://www.microsoft.com/en-us/quantum/technology

Pag. 32 Intro Quantum Computing QRNG Conclusions Summary

1 Introduction

2 Quantum Computing Circuit model Alternatives to circuit model Implementations

3 Quantum Random Number Generators

4 Conclusions

Pag. 33 Intro Quantum Computing QRNG Conclusions What is a random number

A random number is a number generated by an unpredictable process

Pag. 34 Intro Quantum Computing QRNG Conclusions Why random numbers?

Random numbers are crucial in several applications:

1 Information technology and security (also QKD)

2 Scientiﬁc simulation (meteorology, biology, physics...)

3 Lottery/gaming

Pag. 35 Head or tail?

How random is it?

Intro Quantum Computing QRNG Conclusions Generators based on classical physics

How we generate random numbers?

Pag. 36 Intro Quantum Computing QRNG Conclusions Generators based on classical physics

How we generate random numbers?

Head or tail?

How random is it?

Pag. 36 Intro Quantum Computing QRNG Conclusions Generators based on classical physics

Output depends deterministically from initial conditions

Pag. 37 Intro Quantum Computing QRNG Conclusions Pseudo Random Number Generators

Pseudo-random numbers are generated by a deterministic algorithm that produces a sequence that “resemble” a random sequence

PROS CONS

I simple I period

I fast I not-uniformity

I correlations

Pag. 38 Intro Quantum Computing QRNG Conclusions but....

Von Neumann (1903-1957)

(among the father of information theory

"Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin"

Pag. 39 Intro Quantum Computing QRNG Conclusions Problems of PRNG

V1 12589822 V2 490623226 RANDU V3 682947310 V4 1829558474 used by IBM from ’60 to ’90 V5 535857758 V6 1781505818 31 V7 1571347790 Vk+1 = 65539 · Vk mod 2 V8 1984468970 V9 2059651006 V10 940136250

Pag. 40 Intro Quantum Computing QRNG Conclusions Problems of PRNG

Pag. 40 Intro Quantum Computing QRNG Conclusions Flaws in PRNG!

NSA (National Security Agency) scandal

NSA inserted a “backdoor" in the generator Dual_EC_DRBG certiﬁed by NIST

Dual_EC_DRBG was used in several RSA products. In 2013, RSA ofﬁcially discouraged his clients to use their products with Dual_EC_DRBG.

Pag. 41 What QRNG offer:

I intrinsic randomness of quantum measurements

I outputs not predictable even if the initial state is known

I randomness is not due to ignorance on the initial conditions (like coin tossing)

Intro Quantum Computing QRNG Conclusions Why QRNG?

I RANDOM NUMBERS are needed to encrypt all digital communications (email, social networks) and are essential for QKD

Pag. 42 I outputs not predictable even if the initial state is known

I randomness is not due to ignorance on the initial conditions (like coin tossing)

Intro Quantum Computing QRNG Conclusions Why QRNG?

I RANDOM NUMBERS are needed to encrypt all digital communications (email, social networks) and are essential for QKD

What QRNG offer:

I intrinsic randomness of quantum measurements

Pag. 42 I randomness is not due to ignorance on the initial conditions (like coin tossing)

Intro Quantum Computing QRNG Conclusions Why QRNG?

I RANDOM NUMBERS are needed to encrypt all digital communications (email, social networks) and are essential for QKD

What QRNG offer:

I intrinsic randomness of quantum measurements

I outputs not predictable even if the initial state is known

Pag. 42 Intro Quantum Computing QRNG Conclusions Why QRNG?

I RANDOM NUMBERS are needed to encrypt all digital communications (email, social networks) and are essential for QKD

What QRNG offer:

I intrinsic randomness of quantum measurements

I outputs not predictable even if the initial state is known

I randomness is not due to ignorance on the initial conditions (like coin tossing)

Pag. 42 Intro Quantum Computing QRNG Conclusions QRNG based on vacuum ﬂuctuation

Switch between two conjugate quadratures pˆ and qˆ

X p 2 Pguess(p|E) ≤ c(δq, δp)( P(qk)) k

Pag. 43 D. G. Marangon, G. Vallone, P. Villoresi, Phys. Rev. Lett. 118, 060503 (2017) Intro Quantum Computing QRNG Conclusions Randomness estimation

Finite-size min-entropy evaluation

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1.25 GSamples/s, 8-bit resolution Secure bit generation rate of approximately 1.76 Gbit/s (with 5-bit ADC resolution sampling).

Pag. 44 D. G. Marangon, G. Vallone, P. Villoresi, Phys. Rev. Lett. 118, 060503 (2017) Intro Quantum Computing QRNG Conclusions Summary

1 Introduction

2 Quantum Computing Circuit model Alternatives to circuit model Implementations

3 Quantum Random Number Generators

4 Conclusions

Pag. 45 Intro Quantum Computing QRNG Conclusions Conclusions

I Clear trend in quantum computing: from research groups to large companies

I Are we close to quantum supremacy?

I QRNG as a fundamental tool for simulations and security application

Pag. 46 THANK YOU FOR YOUR ATTENTION!

email: [email protected]

http://www.dei.unipd.it/∼vallone

http://quantumfuture.dei.unipd.it/