QECC Tutorial Long

7/30/2019 QECC Tutorial Long

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Quantum Error Correction

Daniel Gottesman

Perimeter Institute


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TheClassical

and

QuantumWorlds


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Quantum Errors

A general quantum error is a superoperator:

Ak Ak

Examples of single-qubit errors:

Bit Flip X: X0 = 1, X1 = 0

Phase Flip Z: Z0 = 0, Z1 = -1Complete dephasing: ( + ZZ)/2(decoherence)

Rotation: R0 = 0, R1 = ei1


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Classical Repetition Code

To correct a single bit-flip error for classicaldata, we can use the repetition code:

0 0001 111

If there is a single bit flip error, we can

correct the state by choosing the majority ofthe three bits, e.g. 010 0. When errorsare rare, one error is more likely than two.


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No-Cloning Theorem

There is no device that will copy an unknownquantum state:

0 00, 1 11

0 + 100 + 11(0 + 1)(0 + 1)

By linearity,

(Related to Heisenberg Uncertainty Principle)


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Barriers to Quantum Error

Correction1. Measurement of error destroys

superpositions.

2. No-cloning theorem prevents repetition.

3. Must correct multiple types of errors (e.g., bitflip and phase errors).

4. How can we correct continuous errors anddecoherence?


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Measurement Destroys

Superpositions?Let us apply the classical repetition code to aquantum state to try to correct a single bit flip error:

0 + 1000 + 111

Bit flip error (X) on 2nd qubit:

010 + 101

2nd qubit is now different from 1st and 3rd. Wewish to measure that it is different without

finding its actual value.


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Measure the Error, Not the Data

00

Use this circuit:

Encoded

state

Errorsyndrome

1st bit of error syndrome says whether the first twobits of the state are the same or different.

2nd bit of error syndrome says whether the second

two bits of the state are the same or different.

Ancillaqubits


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Measure the Error, Not the Data

With the information from the error syndrome,we can determine whether there is an error andwhere it is:

E.g., 010 + 101 has syndrome 11, whichmeans the second bit is different. Correct it witha X operation on the second qubit. Note that the

syndrome does not depend on and .

We have learned about the error without learningabout the data, so superpositions are preserved!


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Redundancy, Not Repetition

This encoding does not violate the no-cloningtheorem:

0 + 1000 + 111

(0 + 1)3

We have repeated the state only in the

computational basis; superposition states arespread out (redundant encoding), but notrepeated (which would violate no-cloning).


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Update on the Problems

1. Measurement of error destroyssuperpositions.





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Correcting Just Phase Errors

Hadamard transform H exchanges bit flipand phase errors:

H (0 + 1) = + + -

X+ = +, X- = -- (acts like phase flip)

Z+ = -, Z- = + (acts like bit flip)

Repetition code corrects a bit flip errorRepetition code in Hadamardbasis corrects a phase error.

+ + -+++ + ---


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Nine-Qubit Code

To correct both bit flips and phase flips, use bothcodes at once:

0 + 1

(000 + 111)3 + (000 - 111)3

Repetition 000, 111 corrects a bit flip error,

repetition of phase +++, --- corrects a phase errorActually, this code corrects a bit flip and aphase, so it also corrects a Y error:

Y = iXZ: Y0 = i1, Y1 = -i0(global phase

irrelevant)


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Update on the Problems






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Correcting Continuous Rotation

Let us rewrite continuous rotation

R0 = 0, R1 = ei1

R = ( )= ei/2( )= cos (/2) I - i sin (/2) Z

1 00 ei

e-i/2

00 ei/2

R(k) = cos (/2) - i sin (/2) Z(k)

(R(k) is R acting on the kth qubit.)


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Correcting Continuous Rotations

How does error correction affect a state with acontinuous rotation on it?

R

(k) = cos (/2) - i sin (/2) Z(k)

cos (/2)I - i sin (/2) Z(k)Z(k)

Error syndrome

Measuring the error syndrome collapses the state:Prob. cos2 (/2): (no correction needed)

Prob. sin2 (/2): Z(k) (corrected with Z(k))


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Correcting All Single-Qubit Errors

Theorem:If a quantum error-correcting code (QECC)corrects errors A and B, it also corrects A + B.

Any 2x2 matrix can be written as I + X + Y + Z.

A general single-qubit error Ak Ak acts like amixture of Ak, and Ak is a 2x2 matrix.

Any QECC that corrects the single-qubit errors X, Y,and Z (plus I) corrects every single-qubit error.

Correcting all t-qubit X, Y, Z on t qubits (plus I)

corrects all t-qubit errors.


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The Pauli Group

Define the Pauli group Pn on n qubits to begenerated by X, Y, and Z on individual qubits.Then Pn consists of all tensor products of up to n

operators I, X, Y, or Z with overall phase 1, i.

Any pair M, N of Pauli operators either commutes(MN = NM) or anticommutes (MN = -NM).

The weight of M Pn is the number of qubits onwhich M acts as a non-identity operator.

The Pauli group spans the set of all n-qubit errors.


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Small Error on Every Qubit

Suppose we have a small error U on everyqubit in the QECC, where U = I + E.

Un = + (E(1) + ... + E(n)) + O(2).

Then

If the code corrects one-qubit errors, it correctsthe sum of the E(i)s. Therefore it corrects theO() term, and the state remains correct toorder 2.

A code correcting t errors keeps the state

correct to order t+1.


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QECC is Possible






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QECC Conditions

Thm.: A QECC can correct a set Eof errors iff

iEaEbj = Cabij

where {i} form a basis for the codewords,and Ea, EbE.

Note:The matrix Cab

does not depend on i and j.

As an example, consider Cab = ab. Then we canmake a measurement to determine the error.

If Cab has rank < maximum, code is degenerate.


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Erasure Errors

Suppose we know the location of an error, but notits type (I, X, Y, or Z). This is called an erasure.

By QECC conditions iEa

Ebj = Cabij:Correct t general errors Correct 2t erasures

For erasures, QECC conditions become:

TrA does not depend on encoded state ,where A is set of qubits which are not erased.

That is, erased qubits have no info. about .


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Stabilizer Codes


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Error Syndromes Revisited

Let us examine more closely the error syndromefor the classical repetition code.

For correctly-encoded state 000 or 111: first

two bits have even parity (an even number of1s), and similarly for the 2nd and 3rd bits.

We can rephrase this by saying a codeword is a+1 eigenvector of ZZI and a state with an erroron the 1st or 2nd bit is a -1 eigenvector of ZZI.

For state with erroron one of the first two bits:

odd parity for the first two bits.


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Error Syndromes Revisited

For the three-qubit phase error correcting code, acodeword has eigenvalue +1 for XXI, whereasa state with a phase error on one of the first two

qubits has eigenvalue -1 for XXI.

Measuring ZZ detects bit flip (X) errors, andmeasuring XX detects phase (Z) errors.

Error syndrome is formed by measuring enoughoperators to determine location of error.


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Stabilizer for Nine-Qubit Code

Z Z

Z Z

Z Z

Z Z

Z Z

Z Z

X X X X X XX X X X X X

M1

M2

M3

M4

M5M6

M7M8

We can writedown all the

operatorsdeterminingthe syndromefor the nine-

qubit code.

These generate a group, the stabilizer of the code,consisting of all Pauli operators M with the property

that M = for all encoded states .


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Properties of a StabilizerThe stabilizer is a group:

If M = and N = , then MN = .

The stabilizer is Abelian:

If M = and N = , then

(MN-NM) = MN - NM = 0(For Pauli matrices)

MN = NMGiven any Abelian group S of Pauli operators, definea code spaceT(S) = { s.t. M = M S}.Then T(S) encodes k logical qubits in n physical

qubits when S has n-k generators (so size 2n-k).


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Stabilizer Elements Detect Errors

Suppose M S and Pauli error E anticommuteswith M. Then:

M (E) = - EM = - E,

so E has eigenvalue -1 for M.

Conversely, if M and E commute for all M S,

M (E) = EM = E M S,so E has eigenvalue +1 for all M in the stabilizer.

The eigenvalue of an operator M from the stabilizer

detects errors which anticommute with M.


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Distance of a Stabilizer Code

Let S be a stabilizer, and let T(S) be thecorresponding QECC. Define

N(S) = {N Pn s.t. MN=NM M S}.Then the distance d of T(S) is the weight of thesmallest Pauli operator N in N(S) \ S.

The code detects any error not in N(S) \ S (i.e., errorswhich commute with the stabilizer are not detected).

Why minus S? Errors in S leave all codewords

fixed, so are not really errors. (Degenerate QECC.)


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Error Syndromes and Stabilizers

To correct errors, we must accumulate enoughinformation about the error to figure out whichone occurred.

The error syndrome is the list of eigenvalues ofthe generators of S: If the error E commuteswith M S, then M has eigenvalue +1; if E andM anticommute, M has eigenvalue -1.

We can then correct a set of possible errors ifthey all have distinct error syndromes.


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Stabilizer Codes Correct Errors

A stabilizer code with distance d corrects (d-1)/2

errors (i.e., to correct t errors, we need d = 2t+1):

E and F act the same, so we need not distinguish.

Thm.: The code corrects errors for which EF N(S) \ S for all possible pairs of errors (E, F).

E and F have same error syndrome

E and F commute with same elements of S

EF N(S)

EF S EF = F = E


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Stabilizer Codes Summary

Choose an Abelian subgroup of the Pauligroup. This will be the stabilizer S of the QECC.

The codewords: { s.t. M = M S}

If S has r generators on n qubits, the QECChas k = n-r encoded qubits.

The codes corrects errors if EF N(S) \ S forall pairs (E, F) of possible errors. The distanced is the minimum weight of N(S) \ S.


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Application: 5-Qubit Code

We can generate good codes by picking anappropriate stabilizer. For instance:

X Z Z X I

I X Z Z X

X I X Z Z

Z X I X Z

n = 5 physical qubits

- 4 generators of S

k = 1 encoded qubit

Distance d of this code is 3.

Notation: [[n,k,d]] for a QECC encoding k logicalqubits in n physical qubits with distance d. The

five-qubit code is a non-degenerate[[5,1,3]] QECC.


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The Smallest QECC

Can we do better than a [[5,1,3]] QECC? No.

Recall that correcting 1 general error isequivalent to correcting 2 erasure errors.

Suppose we had a 4-qubit code whichcould correct 2 erasure errors:

Each pair of qubitshas 2 erasures

Cloning machine


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Classical Linear Codes

A large and useful family of classical error-correcting codes can be defined similarly, usinga parity check matrix. Let H be a (n-k) x nbinary matrix, and define a classical error-

correcting code C of n-bit vectors by

v C Hv = 0.

C is linear: v,w C v+w C. Also, let the

distance d of C be the weight (# of non-zeroentries) of the smallest non-zero v C. Then acode with distance 2t+1 corrects t errors: theerror syndrome of error e is He, and He = Hf

only if e+f C.


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Classical Hamming Codes

H =

( )1 1 1 1 0 0 01 1 0 0 1 1 01 0 1 0 1 0 1

Define a parity check matrix whose columnsare all vectors of length r. E.g., for r=3:

This code has distance 3: if error e has weight1, the error syndrome He specifies its location.

Thus, the Hamming code for r is an [n=2r

-1,k=2r-r-1, d=3] ECC (with k logical bits encodedin n physical bits and distance 3).

E.g., for r=3, we have a [7,4,3] code.


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Linear Codes and Stabilizers

The classical parity check matrix H is analogousto the stabilizer S of a quantum error-correctingcode. Indeed, if we replace all of the 1s of H

with Z operators, we get a stabilizer S definingexactly the same classical code. In particular, itcan correct the same number of bit-flip errors.

E.g., Stabilizer of the [7,4,3] Hamming code:Z Z Z Z I I IZ Z I I Z Z I

Z I Z I Z I Z


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CSS CodesWe can then define a quantum error-correctingcode by choosing two classical linear codes C1and C2, and replacing the parity check matrix of C1with Zs and the parity check matrix of C

2with Xs.

X X X X I I IX X I I X X IX I X I X I X

Z Z Z Z I I IZ Z I I Z Z I

Z I Z I Z I Z

E.g.:C1: [7,4,3]Hamming

C2: [7,4,3]Hamming

[[7,1,3]]QECC


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Which CSS Codes Are Possible?

Not all pairs C1 and C2 are possible: thestabilizer must be Abelian.

If v C1 and w C2, the corresponding Paulioperators commute iff v w = 0. Thus, w C2is also in (C1

) = C1.

The dual C

of a classical code C is the set ofvectors w s.t. v w = 0 for all v C. The rowsof the parity check matrix for C generate C.

To make a CSS code, we require C2 C1.


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Properties of CSS Codes

The parameters of a CSS code made from C1, a[n,k1,d1] code, and C2, a [n,k2,d2] code, are

[[n, k1 + k2 -n, d]] with d

min (d1,d2).Why ? Because of degeneracy (e.g., 9-qubit code).

Codewords of a CSS code are superpositions

of classical codewords: For v C1,v = v+w

w C2

If v-v C2, v and v are the same state, so v

should run over C1/C2. (Recall C2 C1.)


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Summary: Stabilizer Codes

We can describe a quantum stabilizer codeby giving its stabilizer, an Abelian subgroup ofthe Pauli group.

By looking at the stabilizer, we can learn allof the most interesting properties of a QECC,including the set of errors it can correct.

One interesting and useful class of stabilizercodes is the family of CSS codes, derivedfrom two classical codes. The 7-qubit code is

the smallest example.


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Quantum Error Correction Sonnet

We cannot clone, perforce; instead, we splitCoherence to protect it from that wrongThat would destroy our valued quantum bitAnd make our computation take too long.

Correct a flip and phase - that will suffice.

If in our code another error's bred,We simply measure it, then God plays dice,Collapsing it to X or Y or Zed.

We start with noisy seven, nine, or five

And end with perfect one. To better spotThose flaws we must avoid, we first must striveTo find which ones commute and which do not.

With group and eigenstate, we've learned to fixYour quantum errors with our quantum tricks.


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Fault-Tolerant Quantum

Computation


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Error Propagation

When we perform multiple-qubit gates during aquantum computation, any existing errors in thecomputer can propagate to other qubits, even if

the gate itself is perfect.CNOT propagatesbit flips forward:

0 1

0 0

0

1

1

Phase errorspropagate backwards:

0 + 1

0 + 1 0 + 1

0 + 1

0 - 1

0 - 1

0 - 1


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Fault-Tolerance

ECFault-toleranterror correction

Fault-tolerantmeasurement

We need:

Fault-tolerantencoded gates

Fault-tolerantpreparation ofencoded state

We need to avoid error propagation, and also needto perform gates without decoding the QECC.


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Ideal Decoder and t-Filter

Ideal decoder

Corrects errors anddecodes state producingan unencoded qubit.

These operations cannot be performed using realgates, but they are useful for defining and provingfault-tolerance.

tt-Error filter

Projects on states thatare within t errors of avalid codeword

code block decoded qubit


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Properties of FT gates

GateProp. 1:

=

(decoded)

Gate

Prop. 2:

=r r+s

Errors propagate benignly. (Similarly for preparation.)

A good FT gate performs the right gate on the encodedstate. (Similarly for preparation and measurement.)

total per

block

total total

s r s

# errors

r s

(r+s 1 for distance 3 code)


r

total


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Transversal Operations

Error propagation is only a serious problem if itpropagates errors within a block of the QECC. Thenone wrong gate could cause the whole block to fail.

The solution: Perform gates transversally - i.e. onlybetween corresponding qubits in separate blocks.

7-qubit code

7-qubit code


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Encoded X and Z

X X X X I I IX X I I X X IX I X I X I X

Z Z Z Z I I IZ Z I I Z Z IZ I Z I Z I Z

Operations which commute with a stabilizer, but

are not in it, perform encoded operations: We canidentify them as logical Pauli gates. Also, theyare transversal.

X X X X X X X

Z Z Z Z Z Z Z

X

Z


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Logical Clifford Group GatesFor the 7-qubit code, CNOT, Hadamard H, andphase gate P = R/2 = diag (1,i) can all be donewith transversal gates. (They generate the

Clifford group, which can be efficiently simulatedclassically.)

For instance, for CNOT:

CNOT7v+wv+w= v+w(v+v)+(w+w)

= (logical CNOT) vv


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FT Preparation and Measurement

FT measurement for a CSS code:

v = v+ww C2

If we measure each qubit, we get a classicalcodeword v from C1, shifted by a randomcodeword from C2

. Use classical EC to find v.

FT preparation:

1. Encode using any circuit.2. Perform FT error correction.

3. Test encoded state.


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Properties of FT Error Correction

ECProp. 1:EC = EC s

Good error correction corrects any pre-existing errors.

EC =

Good error correction leaves the encoded state alone.

ss

(s 1 for distance 3 code)


ECProp. 2:

s rr


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Fault-Tolerant Error Correction

How can we perform error correction transversally?

0

Encodedstate

Non-fault-tolerant measurement of Z Z Z Z:

An error here couldpropagate to two qubits.


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How can we perform error correction transversally?

Even strings

0000 +1111

Encodedstate

Fault-tolerant measurement of Z Z Z Z:

H

H

H

H

Measureparity


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Create and verify cat states 0000 + 1111.

Measure stabilizer generators to learn syndrome.

Repeat for more confidence.

More advanced FT syndrome measurementtechniques use more complicated ancillas, but

fewer operations on the data:

Steane error correction (uses encoded states)

Knill error correction (based on teleportation)

Shor fault-tolerant error correction:


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Steane Error Correction

data

block

0 + 1

For CSS codes:

H0

Ancilla blocks are encoded using same code(verify them before using)

bit flips propagateforwards

phase errorspropagate backwards

Perrors inclassicalcodewordsreveal

locations oferrors

random classicalcodeword w/ error


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Universal Fault-Tolerant QC

To complete a universal set of gates, we need someadditional gate outside the Clifford group, forinstance the /8-gate: R/40 = e-i/80, R/41 =

ei/81.We cannot perform it transversally, so we will needa more complicated construction:

Create special ancilla state Perform teleportation-like procedure

Perform Clifford group logical correction


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Teleporting Quantum Gates

Quantum teleportation followed by U:

Bell

measurement

U

2 classical bits

Q U

Pauli operator Qspecified by classicalbits

UQUU

Correction operator UQUspecified by classical bits

00 +11

Specialancilla


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Fault-Tolerant /8-Gate/8 gate has a special and useful property:

R/4 X R/4 = e-i/4 PX, R/4 Z R/4

= Z

The correction required after teleportation is aClifford group gate, for which we already knowa fault-tolerant procedure!

Create ancilla state: encoded 00 + ei/4

11 Perform teleportation-like procedure

Perform Clifford group logical correction


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Extended Rectangles

Definition:An extended rectangle (or ExRec)consists of an EC step (leading), followed by an

encoded gate, followed by another EC step (trailing).

Definition:An ExRec is good if it contains at most one

error (roughly speaking) in a gate in the ExRec.

Note: Extended rectangles overlap with each other.

EC EC EC

1st ExRec 2nd ExRec


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Good Circuits are Correct

Lemma [ExRec-Cor]:An ideal decoder can be pulledback through a good ExRec to just after the leading EC.

EC EC =

EC =

(gate ExRec)

(preparation ExRec)

EC = EC

(measurement ExRec)

EC


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Correct Means What It Ought To

Suppose we have a circuit consisting of only goodExRecs. Then its action is equivalent to that of thecorresponding ideal circuit:

EC EC EC

1. Use ExRec-Cor for measurement to introduce an

ideal decoder before the final measurement.2. Use ExRec-Cor for gates to push the ideal decoder

back to just after the very first EC step.

3. Use ExRec-Cor for preparation to eliminate the

decoder.


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Fault-Tolerant SimulationA fault-tolerant circuit simulates the behaviorof a circuit on the encoded qubits. We canreplace each ExRec with the correspondinggate. Suppose the ExRec has A gates, and

each gate is wrong with probability p:Good ExRec:

If there are 0 or 1 errors in the ExRec,replace the ExRec with the ideal gate.

Bad ExRec:

With prob. ~(A2/2)p2 the ExRec is bad,and we replace it with a faulty gate.

For small enough p, error rate decreases.


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Summary of Fault Tolerance

We can create fault-tolerant circuits from anystabilizer code, but the 7-qubit code is particularlystraightforward.

We perform many logical gates by transversaloperations to avoid error propagation within blocks.

Not all operations can be performed this way - theother trick is to create special ancillas and workhard to verify them.


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Threshold for Fault Tolerance


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Concatenated Codes

Error correctionis performedmore frequentlyat lower levels ofconcatenation.

Threshold for fault-tolerance proven usingconcatenated error-correcting codes.

Effective error rateOne qubit isencoded as n,which areencoded as n2,

(for a code correcting 1 error)

p Cp2


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History of the Threshold


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History of the ThresholdShor (1996) - FT protocols

Aharonov, Ben-Or

(1996) - threshold

proof

Knill, Laflamme (1996)

storage thresholdKitaev (1996-...)

topological FT,

thresholdK, L, Zurek (1996)

threshold

Zalka (1996)

simulation

Dennis et al. (2001)

topological threshold

Other

simulations

G, Preskill

higher value

Golden

Age (1996)

Renaissance (2004-)

Dark

Ages

Knill (2004),

Reichardt (2004)

very high threshold

Local gates, specific systems, ...

Aliferis, G, Preskill

(2005) - simple proof

Reichardt (2005)

d=3 proof

Th DiVi C it i


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The DiVincenzo Criteria

1. A scalable physical system with well-

characterized qubits.2. The ability to initialize the state of the qubits

to a simple fiducial state, such as 0...0.3. Long relevant decoherence times, much

longer than the gate operation time.

4. A universal set of quantum gates.

5. A qubit-specific measurement capability.

6. The ability to interconvert stationary andflying qubits.

7. The ability to faithfully transmit flying qubitsbetween specified locations.


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Requirements for Fault-Tolerance

1. Low gate error rates.2. Ability to perform operations in parallel.

3. A way of remaining in, or returning to, the

computational Hilbert space.4. A source of fresh initialized qubits during

the computation.

5. Benign error scaling: error rates that do notincrease as the computer gets larger, andno large-scale correlated errors.


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Additional Desiderata

1. Ability to perform gates between distantqubits.

2. Fast and reliable measurement andclassical computation.

3. Little or no error correlation (unless theregisters are linked by a gate).

4. Very low error rates.

5. High parallelism.

6. An ample supply of extra qubits.

7. Even lower error rates.


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Tradeoffs Between Desiderata

It is difficult, perhaps impossible, to find a physical

system which satisfies all desiderata. Therefore, weneed to study tradeoffs:which sets of properties willallow us to perform fault-tolerant protocols? Forinstance, if we only have nearest-neighbor gates, whaterror rate do we need?

The mandatory requirements for fault-tolerance are nottoo strenuous -- many physical systems will satisfy them.

However, we will probably need at least some of the

desiderata in order to actually make a fault-tolerantquantum computer.


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The Meaning of Error Rates

Gate errors: errors caused by an imperfectgate.

Storage errors: errors that occur evenwhen no gate is performed.

Cited error rates are error probabilities;that is, the probability of projecting ontothe correct state after one step.

E.g.: Rotation by angle has errorprobability ~ 2.

Error rates are for a particular universal gate set.


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Determining the Threshold ValueThere are three basic methodologies used todetermine the value of the threshold:

Numerical simulation: Randomly choose errors on acomputer, see how often they cause a problem. Tendsto give high threshold value, but maybe this is anoverestimate; only applies to simple error models.

Rigorous proof: Prove a certain circuit is fault-tolerantfor some error rate. Gives the lowest threshold value,but everything is included (up to proofs assumptions).

Analytic estimate: Guess certain effects are negligibleand calculate the threshold based on that. Gives

intermediate threshold values.


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Threshold Values

Computed threshold value depends on error-correctingcode, fault-tolerant circuitry, analysis technique. Assumefor now that all additional desiderata are satisfied.

Concatenated 7-qubit code, standard circuitry:

Threshold ~ 10-3(various simulations)

Threshold ~ 3 x 10-5(proof: Aliferis,Gottesman, Preskill, quant-ph/0504218; also

Reichardt, quant-ph/0509203) Best known code: 25-qubit Bacon-Shor code

Threshold ~ 2 x 10-4 (proof: Aliferis, Cross,quant-ph/0610063)


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Ancilla FactoriesBest methods trade extra ancilla qubits for error rate:Ancilla factories create complex ancilla states tosubstitute for most gates on the data. Errors onancillas are less serious, since bad ancillas can bediscarded safely (Steane, quant-ph/9611027).

Extreme case: Create all states using error-detectingcodes, ensuring a low basic error rate but very highoverheads (e.g. 106 or more physical qubits perlogical qubit) -- Knill, quant-ph/0404104, Reichardt,

quant-ph/0406025.

Simulations: threshold ~ 1% or higher.

Provable threshold~ 10-3 (Reichardt, quant-ph/0612004,

Aliferis, Gottesman, and Preskill, quant-ph/0703264)


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Parallel Operations

Error correction operations should be applied inparallel, so we can correct all errors beforedecoherence sets in.

Fault-tolerant gates are easily parallelized.

Threshold calculations assume full parallelism.


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Erasure Errors

For instance: loss of atoms

Losing one is not too serious,but losing all is fatal.

Erasures are an issue for:

Quantum cellular automata

Encoded universality


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Fresh Ancilla States

We need a constant source of fresh blankqubits to perform error correction.

Thermodynamically, noise introduces

entropy into the system. Error correctionpumps entropy into cold ancilla states.

a) Used ancillas become noisy.b) Ancillas warm up while they

wait.

Data

Ancilla


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Fresh Ancilla States

Used ancillas can be replaced by newancillas, but we must ensure ancillas do notwait too long: otherwise, there is anexponential loss of purity.

In particular:

It is not sufficient to initialize all qubits

at the start of computation.

For instance, this is a problem for liquid-stateNMR.


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Large-Scale Error Rates

The error rate for a given qubit should notincrease when we add more qubits to thecomputer.

For instance:

Short-range crosstalk is OK, since it stopsincreasing after neighbors are added.

(See Aharonov, Kitaev, Preskill, quant-ph/0510231.)

Long-range crosstalk (such as 1/r2Coulomb coupling)


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Correlated Errors

Small-scale correlations are acceptable:

We can choose an error-correcting code whichcorrects multiple errors.

Large-scale correlations are fatal:

A large fraction of the computer fails withreasonable probability.

Note: This type of error is rare in most weakly-coupled systems.


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Correlated Errors

Small-scale correlations are not fatal, butare still better avoided.

We assume correlated errors can occur whena gate interacts two qubits. Any other sourceof multiple-qubit errors is an additional errorrate not included in the threshold calculations.

The worst case is correlated errors within ablock of the code, but the system can bedesigned so that such qubits are well separated.

G


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Long-Range Gates

Most calculated thresholds assume we canperform gates between qubits at arbitrarydistances.

If not, threshold still exists, but we need

better error rates to get a threshold, since weuse additional gates to move data aroundduring error correction.


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Local Gates

Proof that threshold still exists with local gates: Gottesman,quant-ph/9903099; Aharonov, Ben-Or, quant-ph/9906129.

We are starting to understand the value of thethreshold in this case:

With concatenation, in 2D, proven threshold of ~ 10-5(Svore, Terhal, DiVincenzo, quant-ph/0604090)

Almost 2D, w/ topological codes & cluster states,

simulated threshold of ~ 6 x 10-3 (Raussendorf,Harrington, quant-ph/0610082)

Almost 1D: proven threshold of ~ 10-6 (Stephens,Fowler, Hollenberg, quant-ph/0702201)

F Cl i l P i


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Fast Classical Processing

Fast measurement and classical processing isvery useful for error correction to compute theactual type and location of errors.

We can implement the classical circuit withquantum gates if necessary, but this addsoverhead: the classical circuit must be madeclassically fault-tolerant.

May not matter much for threshold? (Theclassical repetition code is very robust.)(Szkopek et al., quant-ph/0411111.)


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Other Error Models

Coherent errors: Not serious; could addamplitudes instead of probabilities, but this worstcase will not happen in practice (unproven).

Restricted types of errors: Generally nothelpful; tough to design appropriate codes. (Butother control techniques might help here.)

Non-Markovian errors: Allowed; when theenvironment is weakly coupled to the system, atleast for bounded system-bath Hamiltonians.(Terhal, Burkhard, quant-ph/0402104, Aliferis, Gottesman, Preskill,quant-ph/0504218, Aharonov, Kitaev, Preskill, quant-ph/0510231.)


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Summary

Quantum error-correcting codes exist which cancorrect very general types of errors on quantumsystems.

A systematic theory of QECCs allows us to buildmany interesting quantum codes.

Fault-tolerant protocols enable us to accurately

perform quantum computations even if thephysical components are not completely reliable,provided the error rate is below some thresholdvalue.


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Further Information

Short intro. to QECCs: quant-ph/0004072

Short intro. to fault-tolerance: quant-ph/0701112

Chapter 10 of Nielsen and ChuangChapter 7 of John Preskills lecture notes:http://www.theory.caltech.edu/~preskill/ph229

Threshold proof & fault-tolerance: quant-ph/0504218

My Ph.D. thesis: quant-ph/9705052

Complete course on QECCs:http://perimeterinstitute.ca/personal/dgottesman/QECC2007
http://www.theory.caltech.edu/~preskill/ph229http://perimeterinstitute.ca/personal/dgottesman/QECC2007http://perimeterinstitute.ca/personal/dgottesman/QECC2007http://www.theory.caltech.edu/~preskill/ph229


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The Future of Fault-Tolerance

Industrial Age

Experimental FT

Ancilla

factories

Understanding

of resource

tradeoffs

Efficient fault-

tolerance

Large quantum

computers! Quantum

I f ti A

Documents

QECC Tutorial Long