Bulk Spin Resonance Quantum Information Processing

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Bulk Spin Resonance Quantum Information Processing
Yael Maguire Physics and Media Group (Prof. Neil
Gershenfeld) MIT Media Lab
ACAT 2000 Fermi National Accelerator Laboratory,
IL 17-Oct-2000
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Why should we care?

• By 2030 transistor 1 atom, 1 bit 1
  electron, Fab cost GNP of the planet
• Scaling time (1 ns/ft), space (DNA computers ?
  mass of the planet).
• Remaining resource Hilbert Space.

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Bits

• Classical bit
• Analog bit
• Quantum qubit

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More Bits

• 2 Classical Bits
• 2 Quantum Bits

• N Classical Bits
• N binary values
• N Quantum Bits
• 2N complex numbers
• superposition of states
• Hilbert space

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Entanglement

• correlated decay
• project A
• hidden variables?
• action at a distance?
• information travelling back in time?
• alternate universes (many worlds)?
• interconnect in Hilbert space O(2-N) to O(1)

AB
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The Promise

• Examples
• Shors algorithm (1000 bit number)
• O((logN)2?) vs. O(exp(1.923(logN)1/3(loglogN)2/3
  )
• O(1 yr) _at_ 1Hz vs. O(107 yrs) _at_ 1 GFLOP
• Grovers algorithm (8 TB)
• O( ) vs. O(N)
• 27 min. vs. 1 month _at_ same clock speed.

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What do you need to build a quantum computer?

• Pure States
• Coherence
• Universal Family
• Readout
• Projection Operators
• Circuits

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Previous/Current Attempts

• spin chains quantum dots
• isolated magnetic spins trapped ions
• Optical photons cavity QED
• Coherence!
• Breakthroughs
• Bulk thermal NMR quantum computers
• quantum coherent information ? bulk thermal
  ensembles
• Quantum Error Correction
• Correct for errors without observing.
• Add extra qubits ? syndrome

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What do you need to build a quantum computer
using NMR?
Gershenfeld, Chuang, Science (1997) Cory, Havel,
Fahmy, PNAS (1997)

• Pure States
• effective pure states in deviation density matrix
• Coherence
• nuclear spin isolation, 1-10s
• Universal Family
• arbitrary rotations (RF pulses) and C-NOT
  (spin-spin interactions)
• Readout
• Observable magnetization
• Projection Operators
• Change algorithms
• Circuits
• Multiple pulses are gates

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Quantum Mechanics

• wave function
• observables
• pure state
• mixed state
• Hamiltonian (energy)
• evolution
• equilibrium

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Bulk Density Matrix
B0
B1

• 1023 spin degrees of freedom
• rapid tumbling averages inter-molecular
  interactions
• N effective degrees of freedom
• decoherence averages off-diagonal coherences

N spins I (1/2)
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Deviation Density Matrix in NMR
NMR reduced density matrix

• high temperature approximation
• identity can be ignored
• ensemble Fmolecule Fdeviation

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Spin Hamiltonian

• magnetic moment
• angular momentum
• spin precession
• Zeeman splitting
• 2 spin interaction Hamiltonian

A-B
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Magnetic Field and Rotation Operators

• apply a z field
• evolve in field
• two spins, scalar coupling
• evolution 3 commuting operators

Arbitrary single qubit operations
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The Controlled-NOT Gate

• ENDOR (1957)
• electron-nuclear double resonance
• INEPT (1979)
• insensitive nuclei enhanced by polarization
  transfer

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The Controlled-NOT Gate
Input thermal density matrix
CNOT output
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Ground State Preparation

• We want where
• How? Use degrees of freedom to create an
  environment for computational spins.
• 1. Logical Labeling (Gershenfeld, Chuang)
• ancilla spins - submanifolds act as pure states -
  exponential signal
• 2. Spatial Labeling (Cory, Havel, Fahmy)
• field gradients dephase density matrix terms -
  exponential space
• 3. Temporal Labeling (Knill, Chuang, Laflamme)
• use randomization and averaging over set of
  experiments - exponential time

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Algorithms - Grovers Algorithm

• find xn f(xn) 1, f(xm)0
• Initialize L bit registers
• Prepare superposition of states
• Apply operator that rotatesphase by p if f(x)
  1
• Invert about average
• Repeat O(N1/2) times
• Measure state

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NMR Implementation

• Pure state preparation
• Superposition of all statesH RyA(90) RyB(90) -
  RxA(180) RxB(180)
• Conditional sign flip (test for both bits up) C
  RzAB(270) - RzA(90) - RzB(90)
• Invert-about-mean M H - RzAB(90) - RzA(90) -
  RzB(90) - H

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Experimental Implementation of Fast Quantum
Searching, I.L. Chuang, N. Gershenfeld, M.
Kubinec, Physical Review Letters (80), 3408
(1998).
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Quantum Error Correction

• 3-bit phase error correcting code - Cory et al,
  PRL, 81, 2152 (1998) - alanine

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Quantum Simulation

• Feynman/Lloyd - quantum simulations more
  efficient on a quantum computer
• Waugh - average Hamiltonian theory
• Dynamics of truncated quantum harmonic oscillator
  with NMR- Samaroo et al. PRL, 82, 5381.

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Scaling Issues

• Sensitivity vs. System resources
• Decoherence per gate
• Number of qubits

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Scaling
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Scaling
is separable if

• Is it quantum? Schack, Caves, Braunstein,
  Linden, Popescu,
• Initial conditions vs quantumevolution
• But, Boltzmann limit is not scalable

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Polarization Enhancement - Optical Pumping

• Error correction as well (or phonon)

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Decoherence per gate

• Steady state error correction - 10-4 - 10-6

C. Yannoni, M. Sherwood, L. Vandersypen, D.
Miller, M. Kubinec, I. Chuang, Nuclear Magnetic
Resonance Quantum Computing Using Liquid Crystal
Solvents quant-ph/9907063, July 1999
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Number of Qubits

• Seth Lloyd, Science, 261, 1569 (1993) - SIMD CA
• D-A-B-C-A-B-C-A-B-C....
• at worst linear, but may be polylogarithmic
• Shulman, Vazirani (quant-ph/980460) - using SIMD
  CA
• can distill qubits where SNR independent of
  system size

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Our goals

• Develop the instrumentation and algorithms needed
  to manipulate information in natural systems
• Table-Top (size cost)
• investigate scaling issues

500,000
50,000
5,000
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Magnet Design

• Halbach arrays using Nd2Fe14B 1.2T ? 2.0T
• Fermi Lab - iron is a good spatial filter

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Compilation

• Multiplexed Add
• function program madd(cnumif0, cnumif1,
  enabindex, selindex, inputbits, outputbits,
• BOOLlowisleft) outputbits MUST be zeros
• madd.m
• Implements adding a classical number to a
  quantum number, mod 2L.
• If N is the thing we want to factor, then
  selindex says whether N-cnum is less than or
• greater than B N-cnumgtb --gt add cnum, else
  N-cnumltb --gt add cnum - N 2L
• Enabindex must all be 1, else choose the
  classical addend to be zero.
• Edward Boyden, e_at_media.mit.edu
• INPUT
• cnum classical number to be added
• indices column vector of indices on which to
  operate
• carryindex carry qubit that you're using
• L length(outputbits) It's an L-bit adder
  contains L-1 MUXFAs and 1 MUXHA
• if (L!length(inputbits)) MAKE SURE OF THIS!
• program 'Something''s wrong.'
• return

Can you implement? gcc grover.c -o chloroform
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Nature is a Computer
IBM Dr. Isaac Chuang Dr. Nabil Amer MIT
Prof. Neil Gershenfeld Prof. Seth Lloyd U.C.
Berkeley Prof. Alex Pines Dr. Mark
Kubinec Stanford Prof. James Harris Prof.
Yoshi Yamamoto