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Slide1: 

Quantum Computers That Fix Themselves Dave Bacon Caltech Department of Physics Institute for Quantum Information

Slide2: 

The Enemy?

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57 Nobel Prizes 27 Nobel Prizes 0.01 Nobel Prizes / Capita 0.005 Nobel Prizes / Capita Caltech MIT “lies, damn lies, and statistics.” – Leonard Henry Courtney Quantum Computing Mecca Quantum Computing Mecca The California Tech

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Castor canadenis Caltech

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The Quantum Computing Roadmap NIST Boulder Ions

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Have We Learned Anything? 1941 – First programmable electronic calculator. mechanical relays. 1943 – ENIAC. 18000 vacuum tubes: “nature abhors the vacuum tube.” 1947 – Bell labs develops the transistor. 1952 – G. W. Dummer proposes manufacturing electronic equipment in one block with no connecting wires. 1959 – Texas Instruments and Fairchild Semiconductor invent the integrated circuit. 1964 – first IBM 360 series (the concept of an “architecture”) 1964 – Integrated circuit which cost $1000 in 1959 now costs $10. Moore describes his law. 1972 – Intel 8008 and 8 bit microprocessor. 1975 – First personal computer the Altair. 1982 – IBM PC introduced. Personal computer revolution begins.

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History Only Teaches Me Physics ordered regular reducible turbulent complex not reducible Interacting Isolated What happens when many units (defined isolated) interact? Computers are strange beasts in this world: built from parts which order and are regular reductionist’s dream come true yet: complexity of computer running a program is high algorithmic complexity implies reduction cannot be compressed beyond running the algorithm Platyputor

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The Physics Guarantee What is the phase of matter corresponding to the computer? PRACTICAL QUESTION (as opposed to philosophical) There are distinct PHYSICAL reasons why robust classical computation is possible. not all physical systems are equally good for computation: there exists systems whose PHYSICS guarantees their ability to enact robust classical computation. This talk: (1) this point of view for classical computers (2) attempts to port these ideas to a quantum computer Are there (or can we engineer) physical systems whose PHYSICS guarantees robust quantum computation? Rant mode ON

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Two Paths Coding: majority vote of current Transistor Hard Drive Error correction: amplification Coding: majority vote of magnetism Error correction: local energy minimization Fault-tolerant: macro applied field. Fault-tolerant: conducting / insulating phase transition More than storage?

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A Dynamic Ising Model Ising model Dynamic Ising model

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Reminder two-level system radiatively coupled to a thermal reservoir: “heating” “cooling” “non-dissipative”

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A Dynamic Ising Model Ising model Dynamic Ising model (Metropolis update)

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Storage and Manipulation Compare 1D and 2D Ising models at different temperatures for a) storage of information b) manipulation of information Storage in thermal state (infinite time - ensemble is thermal) 1D Ising 2D Ising 1st attempt: Criticism: (1) thermodynamic limit taken (N infinite) (2) what if relaxation to equilibrium takes a long time?

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2nd attempt: 1D Ising Relaxation to thermal Imperfect preparation Storage 2D Ising

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Manipulation Flipping spins is imperfect 1D Ising 2D Ising 1D Ising temperature less than gap manipulation error must be small 2D Ising temperature less than critical temp self correcting

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Self Fixing

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Two Paths Coding: majority vote of current Transistor Hard Drive Error correction: amplification Coding: majority vote of magnetism Error correction: local energy minimization Fault-tolerant: macro applied field. Fault-tolerant: conducting / insulating phase transition More than storage?

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No Quantum Transistor Quantum Cloning Machine “A single quantum cannot be cloned,” Wootters and Zurek, Nature, 1982

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The Quantum Hard Drive? (Kitaev) Do there exist (or can we engineer) quantum systems whose physics guarantees fault-tolerant quantum computation? 1. Coherence preserving. 2. Accessible Fault-Tolerant Operations 3. Universality “self-correcting”

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Old School Error Fix “cold” ancilla “hot” ancilla If criteria holds, then fixing procedure possible prepare out compute Competition between errors and cooling Quantum error correcting code Error correcting criteria:

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Into The Fray Strongly Interacting Many Body Quantum System Thermodynamic limit guarantees robustness Hamiltonian energy eigenstates Barnes and Warren, PRL 85, 856 (2000) [but is only “classical”]

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Quantum Error Correcting Order Parameters Encoded quantum information The quantum information is still here. How do we see it? diagnose fix measure encoded info j

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A Simple Start [4,2,2] quantum error detecting code: “To be an Error and to be Cast out is Part of [Nature’s] Design.” -William Blake

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Supercoherence ZZ ZZ XX XX 1 2 3 4 4 qubit system System Hamiltonian:

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Supercoherence ALL single qubit errors take degenerate ground state to higher energy levels. Single qubit errors change value of (S1,S2) and hence take ground state to higher energy level. FORBIDDEN D. Bacon, Ph.D. thesis, U.C. Berkeley, 2001

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system bath ~ HI = perturbative system-bath coupling HI energy small compared to Esys and Ebath implies decoherence dominated by pathways that conserve unperturbed energies (essentially rotating wave approximation) “cooling” “heating” “non-dissipative” Making Decoherence Work Hard

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two-level atom radiatively coupled to a thermal reservoir: “heating” “cooling” “non-dissipative” At low bath temperatures, heating disappears…….. Supercoherence made all quantum errors “heating”!

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1 2 3 4 Four spin systems with equal strength exchange interactions between all spins: In a Few Years Time… Supercoherent qubit single P atoms in Silicon exchange controlled by voltage mediated barriers Bacon, Brown, and Whaley, PRL 87, 247902 (2001) DiVincenzo, Bacon, Kempe, Burkard, and Whaley, Nature 408, 399 (2001) See: Hellberg, quant-ph/0304150 (2003)

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Supercoherence As the First Step To Achieving the Threshold For Fault-Tolerant Quantum Computation?

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Mysterious Extra Slide

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Ladder Qubits 1 2 i-1 i i+1 n n-1 n-2 n-3 1 2 qubit (i,2) qubit (i,1) The Hamiltonian Z Z Z Z X X X X X X X X X X X X D. Bacon, Ph.D. thesis, U.C. Berkeley, 2001

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Basis Change i i+1 1 2 Z Z Z Z X X X X X X X X X X X X New basis:

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Everybody’s Favorite Ising + transverse field transverse field only Ising + transverse field transverse field only

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Grounded X X k 1 2 Z Z Z Ground state is two-fold degenerate

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Detect and Correct Z Z Z Z X X X X X X X X X X X X

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Into the 2nd Dimension ZZ ZZ XX XX ZZ ZZ ZZ ZZ XX XX XX XX i i+1 i-1 j-1 j j+1

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ZZ ZZ XX XX XX Tic-Tac-“DOH!” XX ZZ ZZ ZZ XX XX ZZ ZZ ZZ ZZ XX XX XX XX

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Subsystem Codes

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Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z X X X X X X X X X X X X X X

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Into the Third Dimension YY YY ZZ ZZ YY XX XX XX XX ZZ ZZ YY YY XX ZZ

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Logical Operators

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Errors ZZ ZZ YY YY ZZ ZZ ZZ ZZ YY YY YY YY i i+1 i-1 j-1 j j+1

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Wherefore Art Toric Codes?

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Summary Learn from our past! Let physics do the work! Towards self-correcting quantum memories Supercoherence as first step?

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Fin Error correcting codes classical quantum Fault-tolerance local structures giving rise to global robustness (CA) Representation theory identification of qubits and logical operations Geometry, Topology? Statistical Physics theory of strongly interacting quantum systems Real Physics physical implementation optical lattices, donor electrons, superconducting circuits, etc. Naturally Fault-Tolerant Quantum Computation

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