Making quantum computers fault tolerantbreichar/talks/Making...Make fault-tolerant a circuit consisting of a universal set of operations, some faulty: 0 1,, Perfect op’s: Faulty

Making quantum computers fault tolerant

Ben ReichardtCaltech

Ancillas

Ancilla

Factory

Ancillas

Ancillas

Enco

ded

linea

r ga

tes

for te

lepo

rtat

ion

Quantum data

Encoded noisy “magic” states

Encoded Toffoli

nonlinear gates for

teleportation

Universal

Distillery

Data

Ancilla factories Universaldistillery

• Place firmer theoretical foundations under promising schemes for quantum computing in the presence of noise

• Cryptography

• Breaks RSA public-key cryptosystem

• Gives unconditionally secure key distribution

Motivations for quantum information processing

• Quantum computing (QC)

• Extended Church-Turing Thesis: Anything physically efficiently computable can be computed efficiently on my laptop

• QC: Extended Church-Turing Thesis is false; there are exponentially-faster algorithms (for interesting problems) by using quantum mechanics

• Simulation & modeling

• for quantum devices,

• chemistry,

• materials (high-T superconductors, new states of matter?)

• Quantum sensing

• Precise measurement and lithography

• Atomic clocks

• Basic science

• Investigate measurement/decoherence, quantum/classical boundary

• Test qu. mechanics on new scales

(but no free lunch…)

• “Qubit”:

• State of n qubits = unit vector in C2n

Quantum information

0 1

|α0|2 + |α1|2 = 1

(α0

α1

)= α0|0〉 + α1|1〉

(αx)x∈0,1n

• “Qubit”:

• State of n qubits = unit vector in C2n

• Computation by local gates, rotate the state vector

Quantum information

0 1

(α0

α1

)= α0|0〉 + α1|1〉

|α0|2 + |α1|2 = 1(αx)x∈0,1n

(α′x)x∈0,1n

• Observing/measuring system collapses it to a single classical bitstring x

• No exponential parallelism

• Have to “finesse” the quantum system to output the classical information you want

Classical information processing

Quantum information processing

• Classical state is a vector of probabilities:

• Valid operations are stochastic maps

pxx∈0,1n px ≥ 0∑

x

px = 1

αxx∈0,1n

∑

x

|αx|2 = 1

1

1

10

1

1

10

-1

-1

-1

• More details:

• Locality (which unitaries are efficient)

• Measuring the system gives outcome x with probability

• (no exponential parallelism, any more than classically, finesse is needed)

• (quantum information processing generalizes the classical model)

• Quantum state is also a vector

• Valid operations are rotations (unitaries)

Q.C. not just

classical computing with

tiny transistors, uses the laws

of quantum mechanics

The universe is quantum

(we think).

l1 norm

l2 norm

The universe is quantum mechanical but it looks classical because of noise…

Exponential

speedups

Polynomial

Quantum algorithms

Approx. Jones polynomial

Simulation

...of dynamics of physical quantum systems

Approx. Jones polynomial

Search &

random walk

Graph traversal

Grover search

Element distinctness

Game tree

evaluation

Fourier sampling & Hidden subgroup

Factoring Discrete log

Graph isomorphism???

Pell's equation

Abelian, some nonabelian HSPs

Quantum cryptography

CS

CS+Physics

Today:

New algorithmic

approach based

on span programs

• Ion traps

• can trap and cool 16-18 qubits

• can entangle 6-8 qubits in a trap

• microfabrication of trap arrays on chips, dealing with increased noise

• in next 2-3 years may be able to compute with 40-60 qubits

• challenges: controlling thousands of traps with dozens of detection channels and lasers along the surface of the chip…

• Superconducting qubits

• 2 qubit local interactions becoming routine

• nonlocal movement& interactions now possible

• noise levels seem promising…

Quantum computing in 2008

• Other technologies:

• Photonic qubits, quantum dots…

[SH

OM

SM'0

5]

• Scaling these systems is a major engineering challenge

• But the basic technologies have been proven, there are intermediate rewards

• And there are no known fundamental difficulties, except…

• Factoring a 2048-bit number uses

• 6 x 1011 gates on

• 10,000 qubits

• Need error 1/1012 per gate

Common obstacle is noise!

!

K-bit number: 72 K3 gates 5 K qubits

versus eK⅓classically

• Physically reasonable noise rates are ~1% error per gate, or maybe 0.1%

∴ Only 100 operations before an error can occur and propagate through the system

• Schrödinger’s cat:

Noise is fundamental problem for quantum computers: entangled systems are fragile

• “Both dead and alive,” in superposition; but collapses to one or the other when observed

• A single stray photon can collapse it — and also analogous states in a quantum computer

• Physically reasonable noise rates are ~1% error per gate, or perhaps 0.1%

live cat

dead cat

( 1√2,

1√2

)

1√2

(|live cat〉+ |dead cat〉)i.e.

1. Engineering

• Not enough— noise is fundamental in quantum systems

2. Fault tolerance

• Enough to engineer the noise rate beneath a constant threshold,

• Then effective noise rate can be decreased arbitrarily (and efficiently) using error-correcting codes

[Von Neumann ’56]

How to deal with noise?

[Von Neumann ‘56]Classical fault tolerance

Make fault-tolerant a circuit consisting of a universal set of operations, some faulty:

0

1 ,,

Perfect op’s:

Faulty op’s: AND, NOT

000

111

001

Transversalgate application

random

permutation

random

permutation

111

000

Error correction

0L =

1L =

Encoding

001

0 1

1

fraction of 1’s

What’s different quantumly?

• Quantum problems:

• Quantum states are continuous, not discrete—need to protect against continuous errors

• No-cloning theorem: Can’t copy a quantum state , so no immediate analog of the repetition code

• But quantum ECCs do exist! [Shor ’95]

0

1

|ψ〉 "→ |ψ〉|ψ〉0 !→ 0n, 1 !→ 1n

E(|ψ〉)

noise

encoded data

Operational def. of QECC

• Quantum problems:

• Quantum states are continuous, not discrete—need to protect against continuous errors

• No-cloning theorem: Can’t copy a quantum state , so no immediate analog of the repetition code

• But quantum ECCs do exist! [Shor ’95] Operationally,

|ψ〉 "→ |ψ〉|ψ〉0 !→ 0n, 1 !→ 1n

|ψ〉 E(|ψ〉)encode

recoverN⊗m(E(|ψ〉)

)

data qubits

➊

ancilla qubits w/ error information

append

ancill

a, copy

errors

inf. int

o it ➋undo errors on encoded data

encoded data

Quantum error-correcting codes exist

• Although quantum states are continuous, correcting a discrete set of errors (bit and phase flips) suffices

• Based on classical linear ECCs: QECC comes from two linear ECCs

( αβ ) !→ ( α

−β )

( αβ ) !→ ( β

α )bit flip error

phase flip error

0

1 (one for bit flips, one for phase flips)

E(|ψ〉)

noise

encoded data

|ψ〉 E(|ψ〉)encode

recoverN⊗m(E(|ψ〉)

)

data qubits

➊

ancilla qubits w/ error information

append

ancill

a, copy

errors

inf. int

o it ➋undo errors on encoded data

encoded data

Quantum error-correcting codes exist

• Although quantum states are continuous, correcting a discrete set of errors (bit and phase flips) suffices

• Based on classical linear ECCs: QECC comes from two linear ECCs

encode

• How can we use these codes?

• Need operations as well as memory

• Error recovery must be resilient to faults during recovery

• How to encode into them in the first place?? (qu problem)

• Over 19% depolarizing error prob. * 4/3 = over 25%

(one for bit flips, one for phase flips)

(0, 0)

cp2 p

p

Prob. diagramfails to commute

Threshold forimprovement: 1/c

Physical error rate p

1 level

2 levels

3 levels

4 levels

Fault-tolerance intuition

!−→

Encodedgate

Error correction

Gate

perfect decoding

perfect decoding

perfect gate

• Compile ideal circuit into “fault-tolerant” (noise-resistant) version, starting with small QECC:

• Concatenate (i.e., repeat) for arbitrarily improved reliability (so arbly long calcs), if starting below a constant noise threshold

• Problem: Noise model at encoded level is not the same as the physical noise model!

Abridged History of Quantum Fault Tolerance

• 1996-97: First fault-tolerance results: QECCs, threshold proofsShor, Steane, Calderbank, Aharonov, Ben-Or, Kitaev, Knill, Laflamme, Zurek, …

• Proved existence of some positive tolerable noise rate using concatenated qu. codes of distance ≥5

• No explicit lower bounds on tolerable noise rate, but estimates were 10-6-10-5 noise per gate

• Moral: Fault tolerance makes quantum computers plausible in the real world

"Dark Ages"-D. Gottesman


• 1997: Aharonov/Ben-Or, Kitaev: Prove positive tolerable noise rate for codes of distance d≥5

Proofs Estimates & simulations

• 2002: Steane: Correct bit flip errors all at once, and then phase flip errors all at once

• based on simulations, estimates 3x10-3 tolerable noise rate per gate

• 2D locality constraint

• Szkopek et al ‘04

• Svore-Terhal-DiVincenzo ‘05

Simulations using distance-3 codes

• Basic estimates:

• Aharonov & Ben-Or ‘97

• Gottesman ‘97

• Knill-Laflamme-Zurek ‘98

• Preskill ‘98

• Optimized estimates:

• Zalka ‘97

• R ‘04

• Svore-Cross-Chuang-Aho ‘05

• 2005: R, Aliferis/Gottesman/Preskill: First explicit numerical threshold lower bounds, threshold for distance-3 codes

Improved threshold result– Modification of standard error correction

scheme increases estimated threshold 3x, to almost 1%.

[R ‘04]





• based on simulations, estimates 3x10-3 tolerable noise rate per gate• 2005: R, Aliferis/Gottesman/Preskill:

First explicit numerical threshold lower bounds, threshold for distance-3 codes

• Postselection

7-bit ancilla prep.

XZ

XZ

XZ

XZ

XZ

XZ

ZX

ZX

ZX

ZX

ZX

ZX

ZX

ZX

ZX

ZX

ZX

ZX

error correction

detection!

(0, 0)

cp2 p

p

Prob. diagramfails to commute

Threshold forimprovement: 1/c

Physical error rate p

1 level

2 levels

3 levels

4 levels

Error-detection-based fault-tolerance

!−→

Encodedgate

Error detection

Gate

perfect decoding

perfect decoding

perfect gate

• Compile ideal circuit into “fault-tolerant” (noise-resistant) version, starting with small QECC: !"#$

• In simulations, tolerates much higher noise rates than error-correction-based FT schemes

• But (previously) no proven positive threshold!

3

example a 3-dimensional lattice ion trap computer withlower-level encoding (like a DFS or code against qubitloss [7]).

Steane’s error correction procedure [5] was developedin a model where memory errors might be significant; heran simulations with a memory error rate from γ/100 toγ. For a more fair comparison, we evaluated several smalloptimizations to his procedure effective in the low mem-ory error rate regime. Steane extracts one syndrome,aborting error correction if that syndrome is trivial. Oth-erwise, he extracts two more syndromes, and looks fortwo agreeing syndromes. We modify this to always ex-tract one syndrome at a time, and to always abort afterretrieving a trivial syndrome – i.e., if the first two syn-dromes agree or if the second syndrome is trivial, don’textract a third one.

B. Simulation method

We largely follow the simulation method and statis-tical procedures as described by Steane [5]. The simu-lator is fast and efficiently scalable since for each bit itonly needs to store either I, X , Y or Z. Although an-cilla states are stabilizer states, there is no need to keeptrack of the stabilizer with a simulator like CHP [8] be-cause the stabilizer is always well known. Even thoughdata blocks may have arbitrary encoded states, there isno need to keep track of the exact, exponentially largequantum state with a simulator like QuIDDPro/D [9],because we only care about the error correction part ofthe circuit.

To increase the credibility of Steane’s simulationmethod, we additionally track errors between 49-bit datablocks. This is important because even blocks which havejust been corrected still have some errors. Every roundwe apply a transverse CNOT gate to or from anotherdata block. We then correct X and Z errors in a randomorder which determines whether the block will be thecontrol or target of the next round’s transverse CNOTgate. To implement this, if a block has not crashed wesave it in a list for later use. As a detail, we only saveblocks after the third round, to allow the distribution oferrors to converge.

C. Results

Figure 4 shows that the physical error rate is reducedslightly. This slight reduction is largely because of thedramatic reduction in the logical error rate (2 or 3 bitphysical errors within a block) shown in Fig. 5.

Figure 6 shows the effect of the new error correctionprocedure on the crash rate compared to Steane’s proce-dure. (A crash is defined as a set of physical errors whichperfect error correction would project to a logical error.)To read off an approximate threshold from the figure, wetake the intersection of the crash rate curve with the line

Physical level 1 ancilla errors

0

.002

.004

.006

.008

.010

.012

.014

.001

.002

.003

.004

.005

.006

.007

.008

.009

.010

Physical error rate

Ph

ys

ica

l a

nc

illa

err

or

rate

Steane Z

Steane X

Reject Z

Reject X

FIG. 4:

Logical level 1 ancilla errors

.000001

.00001

.0001

.001

.01

.1

.001

.002

.003

.004

.005

.006

.007

.008

.009

.010

Physical error rate

Lo

gic

al

an

cil

la e

rro

r ra

te

Steane Z

Steane X

Reject Z

Reject X

FIG. 5:

Effect on threshold

.0001

.001

.01

.1

.002

.003

.004

.005

.006

.007

.008

.009

.010

.011

Physical error rate

Cra

sh

ra

te

Steane

Reject

3/4

FIG. 6:

of slope 3/4 (the rate at which X , Y or Z physical er-rors occur is 3/4 the depolarization rate). This is onlyan approximate result, since the error model at the nexthigher level is not the same as the physical depolarizationerror model. In particular X and Z logical errors are lesscorrelated than X and Z physical errors.

There is a loss of efficiency at higher error rates, but atrates at or below the old threshold, the efficiency is actu-ally just as good or better. See Fig. 7, where preparing aSteane ancilla at zero error rate is defined to take one unit

Effect of postselection in

ancilla preparation[R ‘04]

፼

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B. Simulation method

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C. Results

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Physical level 1 ancilla errors

0

.002

.004

.006

.008

.010

.012

.014

.001

.002

.003

.004

.005

.006

.007

.008

.009

.010

Physical error rate

Ph

ys

ica

l a

nc

illa

err

or

rate

Steane Z

Steane X

Reject Z

Reject X

FIG. 4:

Logical level 1 ancilla errors

.000001

.00001

.0001

.001

.01

.1

.001

.002

.003

.004

.005

.006

.007

.008

.009

.010

Physical error rate

Lo

gic

al

an

cil

la e

rro

r ra

te

Steane Z

Steane X

Reject Z

Reject X

FIG. 5:

Effect on threshold

.0001

.001

.01

.1

.002

.003

.004

.005

.006

.007

.008

.009

.010

.011

Physical error rate

Cra

sh

ra

te

Steane

Reject

3/4

FIG. 6:

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4ʯ״ʯ6ޱ೯౯# #ޱޱ ӹʯffi6ʯ ೯5464ۿʯ״ʯ״״# ๛95೯5ޱ೯5ʯ״״೯5#ޱ೯5ʯ״ #๛ʯ౯״ 54ʯ# ౯$54״ʯ4ޱ# ౯$54ʯʯffi6ʯ 6ޱ೯59៹೯౯౯॰95ޱ೯ۿޱ# # $# ೯ۿ 6״ʯȶ೯״ʯȶ״U᜶4ʯۿU7ʯʯ26״๛ʯ55ʯ״75ʯ೯ ʯ೯ 6౯౯೯೯5ÿʯ״# ʯ״״# ʯY ʯ$5#$ޱ೯5ʯ6״״ 5೯ୱʯ# ʯ9 65

4

Time to prepare encoded ancilla

0

20

40

60

80

100

0

.001

.002

.003

.004

.005

.006

.007

.008

.009

.010

Physical error rate

Steane avg

Reject avg

FIG. 7:

of time. Since the error rate drops rapidly with concate-nation below the threshold, the efficiency loss will showup in practice only at the lowest concatenation levels.

Further optimization could reduce this overhead. Itsexponential dependence on the error rate makes it sen-sitive to small changes to which the threshold itself isinsensitive. For example, we found that modifying the7-bit ancilla verification to only check only three of thefour relevant stabilizers – an idea from [4] – reduced theoverhead by about a factor of four at 1% error rate.

However, the poor efficiency at high error rates proba-bly rules out some generalizations of this ancilla prepara-tion scheme. For example, we expect it will be impracti-cal to apply the scheme to the 343 bit code gotten by con-catenating the Hamming code twice, or to the [[23, 1, 7]]Golay code concatenated with itself.

V. OPTIMIZED ANCILLA PREPARATION

How much more can optimizations of this type gainus? We ran the same simulations with a 49-bit ancillaperfectly prepared, except each bit was subject to a singlepossibility of error, then each block was checked for X and

Z errors. If an error was detected, the entire ancilla wasrejected. For this ideal ancilla, logical errors are muchless correlated than in our preparation procedure.

The simulations showed almost no improvement what-soever in the threshold. Our ancilla preparation proce-dure is already quite well optimized, and the crashes thatstill occur cannot generally be blamed on a faulty ancilla.Hence further optimizations of the error correction pro-cedure for the [[49, 1, 9]] code will need to do more thanimprove ancilla preparation.

We are not able to simulate rejection ancilla prepa-ration for the Golay code concatenated with itself; toomany ancillas must be thrown away for every good an-cilla. (A crude estimate puts the overhead at 1050, al-though optimizations can reduce this enormously.) Weinstead simulated preparation of ideal ancillas as abovefor the Hamming code, with the expectation that resultswould be close to the actual ancilla preparation proce-dure. We found that the threshold increases to about1%. While 1050 is impractically large, it is still a con-stant. Theoretically, it is of interest how much higher thethreshold can be increased by using different codes, moreconcatenation, or improved syndrome interpretation.

VI. CONCLUSION

For quantum computers to become a reality, highlyefficient coding against errors is essential. We have pro-posed an optimized encoded ancilla preparation schemefor the concatenated Hamming code. Simulations showthat the improved scheme decreases the crash rate byalmost two orders of magnitude, and increases the fault-tolerant threshold significantly. There is however a lossof efficiency at higher error rates.

Very recent independent work by Knill [10] uses a sim-ilar idea to increase the threshold. Further research isneeded to make these schemes more practical: still moreefficient, and applicable to more general error models.

[1] Mark S. Byrd, Lian-Ao Wu, and Daniel A. Lidar.Overvew of quantum error prevention and leakage elimi-nation, 2004, quant-ph/0402098.

[2] Dorit Aharonov and Michael Ben-Or. Fault-tolerantquantum computation with constant error rate, 1999,quant-ph/9906129.

[3] John Preskill. Fault-tolerant quantum computa-tion. In Introduction to Quantum Computation, 1998,quant-ph/9712048.

[4] Christof Zalka. Threshold estimate for fault tolerantquantum computation, 1997, quant-ph/9612028.

[5] Andrew M. Steane. Overhead and noise threshold offault-tolerant quantum error correction. Phys. Rev. A,68:042322, 2003, quant-ph/0207119.

[6] Andrew M. Steane. A fast fault-tolerant filter for quan-tum codewords, 2002, quant-ph/0202036.

[7] Jiri Vala. Quantum error correction of a qubit loss, 2004.[8] Scott Aaronson and Daniel Gottesman. Improved simu-

lation of stabilizer circuits, 2004.[9] George F. Viamontes, Igor L. Markov, and John P.

Hayes. Graph-based simulation of quantum computa-tion in the density matrix representation. In Proc. ofQuantum Information and Computation, SPIE Defenseand Security Symp., 2004, quant-ph/0403114.

[10] Emanuel Knill. Fault-tolerant postselected quan-tum computation: threshold analysis, 2004,quant-ph/0404104.

Approximate Effect on Threshold

Improved threshold result– Modification of standard error correction

scheme increases estimated threshold 3x, to almost 1%.

Knill’s threshold result– Estimated 3-6% threshold for

independent depolarizing errors.

[R ‘04]

[Knill ‘04]





• based on simulations, estimates 3x10-3 tolerable noise rate per gate• 2005: R, Aliferis/Gottesman/Preskill:

First explicit numerical threshold lower bounds, threshold for distance-3 codes

• Postselection

• Postselection + Teleportation

|ψ〉

|ψ〉

Alice’s lab

Bob’s lab

time

classical measurement

outcome

Quantum teleportation

prep

are

enta

ngle

d st

ate

measure|ψ〉

|ψ〉

|ψ〉

Applying teleportation to fault tolerance

prep

are

enta

ngle

d st

ate

measure

1. Encoding 2. Decoding 3. Error correction 4. Computation

time

E|ψ〉

E

E|ψ〉

E


enta

ngle

d st

ate

measure

1. Error correction 2. Computation

encoder

* decoding measurements using classical computer

1. Error correction 2. Computation

U |ψ〉

|ψ〉

U


enta

ngle

d st

ate

measure

operation

* operation U should be “C2”

EU |ψ〉

E

E|ψ〉

U Eenta

ngle

d st

ate

measure

time

• Teleportation allows for correcting bit flip errors, phase flip errors, and doing one step of computation all at once.

• (Provided that we can prepare reliably the necessary resource states.)

Error correction + Encoded ComputationApplying teleportation to fault tolerance



• Note: We can prepare very good ancilla states, e.g., throwing away all ancillas with any detected errors (“postselection”). We wouldn’t want to throw away the data—but the data is isolated from the ancilla state.

Teleported EC + encoded computation

EU |ψ〉

E

E|ψ〉

U Eenta

ngl

ed s

tate

measure

time



• Note: We can prepare very good ancilla states, e.g., throwing away all ancillas with any detected errors (“postselection”). We wouldn’t want to throw away the data—but the data is isolated from the ancilla state.

Teleported EC + encoded computation

EU |ψ〉

E

E|ψ〉

U Eenta

ngl

ed s

tate

measure

time

Quantum disadvantages

• States are continuous (i.e., analog)• No-cloning theorem

Quantum advantage!

• Quantum teleportation allows isolating the data from errors

Uncontrolled

50% 50%

Controlled

Uncontrolled

1% 1%

Controlled

Proofs based on controlling events most of the time, with occasional failures

Controlled(well-bounded)

Uncontrolled(worst-case)

99%

1%

Most of the time, errors are detected — but (counterintuitively) survival probability for uncontrolled portion could be much higher

Uncontrolled fraction of probability mass increases exponentially after renormalizing!

Renormalization frustrates previous proofs

• Problem: Although Knill estimated tolerable noise rate was 3-6%, proofs could not show that postselection-based schemes tolerated any noise at all!

• Idea: Maintain inductive invariant of goodness. (A level-k block is good “if it has at most one bad level-(k-1) subblock.”)

• Problems:

• Inefficient analysis: Logical error rate for a distance-d code drops as c p(d-1)/2 instead of c p(d+1)/2

∴ Can’t hope for very good rigorous lower bounds on the noise threshold

• No threshold at all for concatenated d=3 codes, or for postselection-based schemes

ECX

good badXX (one level k-1 error is already too many)

Intuition for Aharonov/Ben-Or’s proof

EC

ECX

X

good

good

good

good

(assuming one level k-1 error, m≥7)

X

X

X

X

EC

ECX

X

good

good

good

bad—uncontrolled!

(two level k-1 errors, m=7)

X

X

X

XX

XX

X

X

XXXXX

• Existence of tolerable noise rates for many fault-tolerance schemes, including:

• Schemes based on error-detecting codes, not just ECCs (Knill-type)

• Distance-3 codes, and more efficient “Fibonacci”-type schemes (d=2 codes)

• Tolerable threshold lower bounds*

• 0.1% simultaneous depolarization noise†

• 1.1%, if error model known exactly

* Subject to minor numerical caveats † Versus .02% best lower bound for error-correction-based FT scheme [Aliferis, Cross 2006]

Results

•!"#$%&'( !−→•

ED••!"#$%&'(

ED!"#$%&'( !"#$%&'(

• Problem: Although Knill estimated tolerable noise rate was 3-6%, proofs could not show that postselection-based schemes tolerated any noise at all!

• Renormalizing the error distribution leads to bad correlations.

[R ’06]

Techniques

• Main new technique is to maintain close control over the distribution of errors in the quantum computer(Previous threshold proofs had used a “worst-case” criterion for error behavior that blew up during renormalization.)

• Rewrite true error distribution as a mixture of nearby distributions whose error distributions lack nasty correlations

true dist.

nice dist.

nice dist.

nice dist.

true dist.

nice dist.

true dist.

nice dist.

nice dist.

nice dist.

• Def: Noisy encoder = perfect encoder, followed by bitwise-indep. noise at rates ≤p.

E

Bitwise-independent noise is nice…

E=

?=

bitwise-independent errors preceding encoded gate

bitwise-independent errors following perfect gate, plus quadratically suppressed

independent logical errors

Induction claim?

ppp

(tool for analysis—such encoders don’t actually exist)

Encodedgate

Error detectionE Perfect

gate Ec p2

(much stronger than Aharonov/Ben-Or’s claim)

1

1

1

1 1

1

1 1

E

E

E

E

=

encoded FT circuit

E

E

E

E

!cp2

!cp2

!cp2

!cp2

!cp2

!cp2

perfect perfect

?=Encodedgate


gate Ec p2

• Since the error model is preserved (level-one logical errors have the same form as physical errors), the analysis can be repeated to give a threshold

⇒

• Def: Noisy encoder = perfect encoder, followed by bitwise-indep. noise at rates ≤p.

E

Bitwise-independent noise is nice…

E=

bitwise-independent errors preceding encoded gate

bitwise-independent errors following perfect gate, plus quadratically suppressed

independent logical errors

Induction claim

ppp

(tool for analysis—such encoders don’t actually exist)

Encodedgate


gate Ec p2

(much stronger than Aharonov/Ben-Or’s claim)

==

• Numerical approach (for numerical threshold lower bounds)

• Existence argument (for threshold existence proofs):

• characterize convex hull of dit-wise independent distributions (a simplex)

• “pull back” actual distribution onto distribution on dits

• Must also obtain universality — CNOT and similar “linear” gates can be efficiently simulated on a classical computer. Need a nonlinear operation (AND or Toffoli). Use “magic states distillation.”

Details in proving that mixing works

coordinate-wise upper & lower bounds on actual error distribution

convex hull of “nice” distributions

• Conclusion: Mixing argument shows that concatenation works to reduce errors. Error events are correlated, but error correlations do not explode.

• Correlations manifest themselves as asymmetries in the conditional error models

—violates a key assumption of Knill, that all gates have symmetrical failure models, at all levels of concatenation

• With postselection, gate error rates that are asymmetrically too low can be just as bad as error rates that are too high

• Are Knill’s simulations too optimistic?

Conclusionstrue dist.

nice dist.

true dist.

nice dist.

nice dist.

nice dist.

true dist.

nice dist.

true dist.

nice dist.

nice dist.

nice dist.

• Magic states distillation (for teleporting into a universal gate set)

• Optimizing ancilla verification

• Higher-order-accurate composite pulses, based on quantum search algorithm, for eliminating noise without encoding

More fault-tolerance work…

[R’05]

[R’06]

[R’05, ‘06]

• Quantum computers need fault-tolerance techniques if they are to scale, but…

• Current FT schemes are not good enough

• Need to increase tolerable noise rate, reduce overhead

• So far, the biggest improvements have come not from optimizations or customizations, but rather from new quantum concepts that unify. The next division to remove is the code concatenation levels.

• Foundations:

• Extend applicability of threshold proofs

• Improve threshold upper bounds

• Connecting full-blown fault-tolerance schemes to implementations

• Specialized, low-level error prevention (e.g., composite pulses, DFSs)

Open questions in fault tolerance

[Farhi, Goldstone, Gutmann ‘07]

Scatter a wave against the tree…

FGG quantum walk |ψt〉 = eiAGt|ψ0〉


ϕ(x) = 0 ϕ(x) = 1

Wave transmits!Wave reflects!


x11 = 0x11 = 1

Farhi, Goldstone, Gutmann ‘07 algorithm

• Theorem ([FGG ‘07, CCJY ‘07]): A balanced binary NAND formula can be evaluated in time N½+o(1).

Questions:

1. Why does it work?

2. How does it connect to what we know already?

3. How does it generalize?

4. What kinds of problems can we hope to solve with this technique?

NAND

NAND

NAND

NAND NAND NAND NAND

x1 x2 x3 x4 x5 x7x6 x8



Questions: Answers:

“span programs” [Karchwer/Wig. ‘93]

formula evaluation problem over extended gate sets







Questions:

“span programs” [Karchwer/Wig. ‘93]

formula evaluation problem over extended gate sets

Answers:





5. No, really, WHY does it work? ???

• Theorem ([FGG ‘07, CCJY ‘07]): A balanced binary AND-OR formula can be evaluated in time N½+o(1).

• Theorem:

• An “approximately balanced” AND-OR formula can be evaluated with O(√N) queries (optimal!).

• A general AND-OR formula can be evaluated with N½+o(1) queries.

unbalanced AND-OR

Analysis by scattering theory.

[FGG ‘07] algorithmNAND

NAND

NAND

NAND NAND NAND NAND

x1 x2 x3 x4 x5 x7x6 x8

[ACRŠZ ‘07] algorithm

Running time is N½+o(1) in each case, after preprocessing.

balanced,more gates

[RŠ ‘08] algorithm• Theorem: A balanced (“adversary-

bound-balanced”) formula φ over a gate set including all three-bit gates (and more…) can be evaluated in O(ADV(φ)) queries (optimal!).

(Some gates, e.g., AND, OR, PARITY, can be unbalanced—but not most!)

• Best quantum lower bound is [LLS‘05]

• Expand majority into AND, OR gates:

∴ AND, OR formula size is ≤ 5d

∴ O(√5d) = O(2.24d)-query algorithm

x1

MAJ MAJMAJ MAJ MAJMAJ

x1 x1

MAJ

MAJ

MAJ

MAJ

MAJ

ϕ(x)

MAJMAJ

...

Recursive 3-bit majority tree

< 7/3 • [Jayram, Kumar & Sivakumar ‘03]

d

3d

MAJ3(x1, x2, x3)= (x1 ∧ x2) ∨ (x3 ∧ (x1 ∨ x2))

Ω(ADV(ϕ) = 2d

)

[ACRSZ ‘07]

• New: O(2d)-query quantum algorithm

[RŠ ‘08] algorithm• Theorem: A balanced (“adversary-

bound-balanced”) formula φ over a gate set including all three-bit gates (and more…) can be evaluated in O(ADV(φ)) queries (optimal!).

(Some gates, e.g., AND, OR, PARITY, can be unbalanced—but not most!)

• Def: An n-bit span program P is*:

• A target vector t in vector space V over C,

• n input subspaces, one for each bit

Span program P computes fP: 0,1n→0,1, fP(x) = 1 ⇔ t lies in the span of subspace i : xi=1

• Ex.: P:

Span program definition

[Karchmer, Wigderson ’93]

x1

x2x3

target t fP = MAJ3

= 1= 1

= 1

* Not the general def.

Weighted bipartite graph

1 0-1 1

1 1

output edge

Span program P

E.g., MAJ3:t =

(10

) (1 0 −11 1 1

)x1

x2x3

target t

= 1= 1

= 1

Matrix

x 1=

1

x 2=

1

x 3=

1

For a given x, add edges above those inputs evaluating to false.

Span program P

E.g., MAJ3:

input edges

t =(

10

)

1 0-1 1

1 1

(1 0 −11 1 1

)x1

x2x3

target t

= 1= 1

= 1

Matrix

x 1=

1

x 2=

1

x 3=

1

Weighted bipartite graph

λ=0 eigenvector computes P

For a given x, add edges above those inputs evaluating to false.

Thm: fP(x) = 1 eigenvalue-0 eigenvector supported on bottom vertex.

⇔ ∃

output edge

x1=1 x2=1 x3=0E.g.,

-1 0

0

0 0

1

1

⇒O(2d)-query (optimal!) recursive MAJ3 evaluation algorithm

Recursive MAJ3 .MAJ


MAJ

MAJ

MAJ

MAJ

MAJ

ϕ(x)

MAJMAJ

• Main Theorem:

• φ(x)=1 AG(x) has λ=0 eigenstate with Ω(1) support on the root.

• φ(x)=0 AG(x) has no eigenvectors overlapping the root with |λ|<1/2d.

⇒

⇒

1 0-1 1

11

aO

bO bC

a1a2 a3

b1 b2 b3

input edges

x1


x1 x1

MAJ

MAJ

MAJ

MAJ

MAJ

ϕ(x)

MAJMAJ

...

Balanced MAJ3

OR

x2x1 xN· · ·

Classical

Θ(N)

AND

OR OR

AND ANDAND AND

x6 x8x7x5x4x2 x3x1

Θ(N0.753…)[S‘85, SW‘86, S‘95]

General read-once AND-OR

Balanced AND-OR

Conj.: Ω(D(f)0.753…) [SW ‘86]

[Nisan ‘91]: R2(f) = Ω(D(f)⅓)

Ω(N0.51) [HW‘91]

Ω((7/3)d), O((2.6537…)d)[JKS ’03]

Quantum

Θ(√N) [Grover ‘96]

...

(fan-in two)

Θ(2d=Nlog32)

and much more…

Θ(√N)

Ω(√N), √N⋅2O(√(log N))

[FGG, ACRŠZ ‘07]

[ACRŠZ ‘07][BS ‘04]

[RŠ ‘08]

NP

PSPACE

Open ?: More quantum algorithms based on span programs?

• Our quantum algorithm evaluates span programs. We’ve applied it by building a large span program by composing small ones for all the gates.

• New framework for developing quantum algorithms: Are there interesting quantum algorithms based directly on large span programs? (E.g., graph problems, Perfect Matching, …) [notion of quantum recursion]

Open:

• Extensions to larger gate sets…

• Unbalaned formulas over more gates…

• Why do span programs work so well? Connection to adversary lower bounds ADV(f) ≤ ADV±(f)?

Bird’s-eye view of quantum computing

What to do with a quantum

computer?

How can we build a quantum

computer?

Computer science

Physics

Adiabatic algorithm even brings these together

Adiabatic

optimizat

ion alg.

Formula e

valuatio

n

Fault tolerance theory

Composite pulses

Documents

Making quantum computers fault tolerantbreichar/talks/Making...Make fault-tolerant a circuit consisting of a universal set of operations, some faulty: 0 1,, Perfect op’s: Faulty