shor’s factoring algorithm and nmr...outline alice dudle, alexia pastré, artemiy burov, tim...

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Artemiy Burov, Alice Dudle, Tim Hofmann, Alexia Pastré

23.04.2018

4/23/2018 1

Shor’s factoring algorithm and NMR

Alice Dudle, Alexia Pastré, Artemiy Burov, Tim Hofmann

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Outline

4/23/2018Alice Dudle, Alexia Pastré, Artemiy Burov, Tim Hofmann 2

I. Introduction

II. Shor’s Algorithm & Main Gate

III. Realization of the gates & Measured spectrum

IV.Results & Conclusion

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What is NMR and how is it used for QIP?


• Nuclear Magnetic Resonance: manipulation and detection of nuclear spin states using

radiofrequency (RF) electromagnetic waves

• Allows to store quantum information in the nuclear spin of atoms in a molecule

• No direct access individual nuclei use of ensemble averages

• EM pulses applied to the sample each molecule responds in roughly the same way

• each molecule ≈ independent computer

sample ≈ huge number of computers all running in parallel (classically)

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DiVincenzo Criteria


1. Qubits: nuclear spins ½ in B0 field (↑ and ↓ as 0 and 1)

2. Ability to initialize the state of the qubits: ”effective pure state preparation”

3. Coherence times: several seconds (but goes down with molecule size scaling problem)

4. Quantum gates: RF pulses

5. Read-out: spin detection with RF coil

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Experimental setting: pulsed NMR system for a liquid sample

kjgkof 7 spin-1/2 nuclei


spectrometer

sample

qubits

support atoms

Perfluorobutadienyl iron complex

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The System Hamiltonian


J (indirect) couplingMultiple spins

in static BDipole-dipole interaction

1 2 3

can be dropped

(averaged away due to rapid tumbling)

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Multiple (uncoupled) spins in static B:

Single spin in static B:1 Zeeman splitting

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J (indirect) coupling: electron-mediated Fermi contact interaction 2

Energy-level diagram

-----: two uncoupled spins

: two spins coupled by HJ

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J (indirect) couplingMultiple spins

in static B

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The Control Hamiltonian


if multiple RF fields

separate rotating frame (at 𝜔0𝑖 ) for each spin i

Iz term dropped out, coupling term remains invariant

Rotating frame

approximation:

Rotating frame Lab frame

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Effective pure state preparation


Effective pure ground state Generates the same signal as

First three qubits

Desired initial state of the 7 qubits :

Temporal

averaging

Thermal equilibrium

Statistical mixture of 0 and 1

Not suitable for quantum computation

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Skeleton of the algorithm

1.12.2014First name Surname (edit via “Insert” > “Header & Footer”) 12

Let N denote the integer to be factorized. Assume that N is not even or a power of a prime (in our

example N = 15, the smallest number for which Shor’s algorithm makes sense).

1. Choose a: 1 < a < N.

2. Compute b = gcd(𝑎, 𝑁). If b > 1 output b.

3. Find the smallest r: 𝑎𝑟 ≡ 1 mod N (Existence – Euler’s theorem). If r is odd, the algorithm has

failed.

4. Compute s = gcd(𝑎 Τ𝑟 2 − 1,𝑁). If s = 1, the algorithm has failed.

5. Output s.

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Step 3 – find the order of r modulo N

• Consider 𝑓: ℤ → ℤ𝑛 defined by 𝑓 𝑥 = 𝑎𝑥mod𝑁

• 𝑓 𝑎 + 𝑏 = 𝑓 𝑎 𝑓(𝑏)

• We require 𝑓 𝑟 = 𝑎𝑟mod𝑁 = 1

• 𝒇 is periodic with period 𝒓, so the problem reduced to finding the period of this function

• For this part, we use quantum computing.

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Step 4 – knowing the order 𝑟, find the divisor of 𝑁

• Write 𝑎𝑟 − 1 as (𝑎𝑟/2−1)(𝑎𝑟/2+1)

• For that 𝑟 has to be even

• 𝑎𝑟/2 − 1 is not divisible by N, we hope that neither is 𝑎𝑟/2 + 1 (probability of that > 1

2)

• Find a divisor of N by computing s = gcd(𝑎 Τ𝑟 2 − 1,𝑁)

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Quantum subroutine


• To find the order 𝑟 efficiently we want to use a quantum computer

• Initialize first register to |0>⊗…⊗|0> and second register to |0>⊗…⊗|1>

• Use Hadamard gate to prepare the first register in uniform superposition of basis states for

parallel computation of 𝑓 𝑥

• Compute 𝑓 𝑥 = 𝑎𝑥 mod 𝑁 for 2𝑛 values in parallel

• Perform inverse QFT (exponentially faster then DFT)

• Measure, the result is those values of x which have the biggest amplitude

Period estimation

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Quantum subroutine


Step 2 – Main gate

• 𝑥𝑛−1, . . , 𝑥0 are the digits of x

• 𝑎𝑥 = 𝑎2𝑛−1𝑥𝑛−1… 𝑎2𝑥1 𝑎𝑥0

• Realized as controlled multiplication by 𝑎2𝑘

• 𝑎2𝑘

precomputed on a classical computer

• We created a superposition of 𝐚𝒎𝒐𝒅 𝑵, 𝒂𝟐 𝒎𝒐𝒅 𝑵,…, 𝒂𝑹 𝒎𝒐𝒅 𝑵 for some big enough R.

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Quantum subroutine


Step 2 – Main gate, N = 15

• Possible values of a are 2,4,7,8,11,13,14

• For a = 2,7,8,13: 𝑎4mod15 = 1

In this case we just need two bits 𝑥0 and 𝑥1

• For a = 4,11,14: 𝑎2mod15 = 1

Then we just need 𝑥0

• In the experiment n = 3 bits are used to store 𝑥 (R = 8) and m = 4 for 𝑓(𝑥)

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Two Qubit Gates


Interaction Hamiltonian:

Evolution Operator

CNOT Gate

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Two Qubit Gates


CNOT Gate:

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Refocusing


Suppressing Interaction if not wanted

Rotation direction changes depending

on control qubit.

Change rotation direction half way

through → no netto rotation

Change rotation direction by appling

gate to the …

… control qubit (a)

… target qubit itself (b)

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Refocusing


Example 4 qubit system

Leaves 1 and 2 coupled

Turns off all other couplings

+ unchanged

- flipped

▌π rotation

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Refocusing


Complex refocusing pulses

for many qubits

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Measured Spectrum


Look at qubit 1

ω = ω0+

𝑖=1

7

±𝐽1𝑖2

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Measured Spectrum


Look at qubit 1

Coupled to qubit 7

ω = ω0+ ±𝐽712

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Measured Spectrum


Look at qubit 1

Coupled to qubit 7 and qubit 4

ω

= ω0+ −𝐽712

±𝐽412

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Measured Spectrum


Look at qubit 1


ω = ω0 + ±𝐽712

±𝐽412

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Measured Spectrum


Look at qubit 1


ω = ω0 + ±𝐽712

±𝐽412

|011

>

|01

0>

|00

1>

|00

0>

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Measured Spectrum


Look at qubit 1

Coupled to all qubits

ω = ω0+

𝑖=1

7

±𝐽1𝑖2

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Measured Spectrum


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• Same coil for RF pulse and for detection

• Only xy-component measured

• Oscillating e.m. field

• Fourier transform: frequency spectrum

• State determined by sign of the peak

Alice Dudle, Alexia Pastré, Artemiy Burov, Tim Hofmann 30

Read-out

4/23/2018

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• Recap :

𝑓 𝑥 = 𝑎𝑥𝑚𝑜𝑑 𝑁Task: find period 𝑟

𝑔. 𝑐. 𝑑. 𝑎 ൗ𝑟 2 ± 1,𝑁

• Easy case: 𝑎 = 11 (𝑁 = 15)• Mixture of 000 and 100

• Periodicity of output: 4

• Period of 𝑓 𝑥 : 𝑟 = Τ2𝑛4 = 2

• G.c.d. 11 Τ𝑟 2 ± 1,15 = 3, 5


Results

4/23/2018

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• Easy case: 𝑎 = 11• Mixture of 000 and 100


• Period: 𝑟 = ൗ234 = 2

• G.c.d. 11 Τ𝑟 2 ± 1,15 = 3, 5

• Difficult case: 𝑎 = 7• Mixture of 000, 010, 100, 110


• Period: 𝑟 = ൗ232 = 4

• G.c.d. 7 Τ𝑟 2 ± 1,15 = 3, 5


Results

4/23/2018

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• Assumption:

Relaxation independent for each spin

• Bloch equations (without precession)

• Spin-lattice relaxation: 𝑑𝑀𝑧

𝑑𝑡= −

1

𝑇1𝑀𝑧 −𝑀0

𝑀𝑧(𝑡) = 𝑀0(1 − 𝑒− ൗ𝑡 𝑇1)

• Spin-spin relaxation: 𝑑𝑀𝑥𝑦

𝑑𝑡= −

1

𝑇2𝑀𝑥𝑦

𝑀𝑥𝑦(𝑡) = 𝑀0𝑒− ൗ𝑡 𝑇2


Decoherence model

4/23/2018

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Positive aspects

• First method to realise quantum

computations

• Some algorithms successfully implemented

• High degree of control

• Development of useful tools for other

quantum computer experiments

Limitations

• Scaling chalenge

• Signal intensity ∝ Τ1 2𝑛

• Coherence time decreases with molecule size

• Difficulty of qubit selection increases with size

• Ensemble average

• Error correction prevented

• Meaningful result ?

• Interactions determined by molecule

No switching off


D. NMR quantum computing: overview

4/23/2018

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• “Experimental realization of Shor’s quantum

factoring experiment using NMR”

L.M.K. Vandersypen & al

Nature, Vol. 414, Dec. 2001

• “NMR techniques for quantum control and

computation”

L.M.K. Vandersypen & I.L Chuang

Review of Modern Physics, Vol. 76, Oct.

2004

• “Quantum computation and quantum

information”

M.A. Nielsen & I.L. Chuang,

Cambridge University Press, 2000

• “Experimental Realization of an Order-

Finding Algorithm with an NMR Quantum

Computer “

L.M.K. Vandersypen & al

Phys. Rev. Let., Vol.85, N°25, Dec. 2000

• “Quantum Computation”

Lecture notes by Ashley Montanaro,

University of Bristol, Spring 2018

• “Magnetic resonance imaging in medicine”

Lecture by Prof. S. Kozerke, ETH & UZH,

Spring 2017

• QIP lectures


References

shor’s factoring algorithm and nmr...outline alice dudle, alexia pastré, artemiy burov, tim...

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