Fault-tolerant classical computing

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Photos placed in horizontal posi1on with even amount of white space between photos and header

Fault-tolerant quantum computing Andrew J. Landahl

12/5/11

QEC11: Second Interna1onal Conference on Quantum Error Correc1on

p

p fail |M〉 • MX

!"#$%&'(∣∣0

⟩/ H • )*+,Unencode -./01234 MZ •

∣∣0⟩

/ -./01234 X Z∣∣M

⟩

! ! ! ! ! ! ! ! ! ! !"""""

"""""

! ! ! ! ! ! ! ! ! ! !

This work was supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories.

Fault-tolerance

QEC11: Second Interna1onal Conference on Quantum Error Correc1on 2

Task: Maintain data. (Sta1c: memory; moving: communica1on.) Failure: Data corrup1on. (Source: local environmental noise.)

Process: Error correc1on. Solves by adding redundancy and processing.

Fault tolerance: Can achieve failure probability ε by an O(log (1/ε))‐sized code as long as p < pc, the error threshold for the infinite code family.

New Problem 1: The processing can itself fail! New Problem 2: How do we compute with data now that it is encoded?

Defini:on: A process is fault‐tolerant if it achieves its task (arbitrarily well, efficiently) in the event of a failure.

Example: (relevant for both classical and quantum informa:on)

A fault‐tolerant compu&ng protocol maintains arbitrary computa1ons (arbitrarily well, efficiently) in the presence of faults to both its data and its processing.

Central ques:ons: •  What combina1on of computa1on, control, and noise models admit fault‐tolerant computa1on?

•  What are the (minimal) resource costs for achieving fault‐tolerant quantum computa1on?

Fault-tolerant classical computing


•  Modern computers can calculate for weeks (years?) without failing: good hardware. •  With faulty hardware: S1ll possible! (Cost, power, speed, etc. may drive us there.)

SeGng: •  Computa:onal model: Circuits in which each gate has exactly one output (formulas).

•  Control model: Parallel opera1on, refreshable bits allowed.

•  Noise model: Noisy gate = ideal gate followed by bit‐flip with probability p.

Approach: •  Simulate an ideal formula to desired precision ε using a faulty formula.

•  Encode data and process it with encoded gates which suppress error spreading.

Threshold Theorem: •  A g‐gate ideal formula can be simulated to precision ε by an O(g log (g/ε))‐gate faulty formula as long as p < pc,

the accuracy threshold for classical computa1on.

•  2‐input gate accuracy threshold (1ght):

•  (2k+1)‐input gate accuracy threshold (1ght):

pc = (3−√

7)/4 ≈ 8.9% [Unger, 2008]

[Evans and Schulman, 2003; Hajek and Weller, 1991]

[von Neumann, 1952, 1956, 1966]

pc =12− 2k−2

k( k−1k/2−1/2

) k→∞−→ 50%


•  Computa:onal model 1.  Quantum Turing machine

2.  Quantum walk

3.  Adiaba1c

4.  Quantum circuit

•  Topological •  Measurement‐based

•  Holonomic

•  Necessary •  Helpful

•  Convenient

•  Realis1c

•  Noise model •  Nonincreasing error rate

•  Reliable classical computa1on

•  No qubit leakage

•  Uncorrelated noise

•  State‐independent noise •  Uniformly faulty gates

•  Control model •  Parallel opera1on

•  Refreshable qubits

•  Fast classical computa1on

•  Finite gate basis

•  Equal‐1me gates •  2D layout

•  Local quantum processing

} The only model proven to admit fault‐tolerant quantum compu:ng

Fault-tolerant quantum computing

H(t = 0)

H(t = τ)

Fault-tolerant quantum computing


Four components of every known fault‐tolerant quantum compu:ng protocol (FTQCP) 1.  An infinite family of (stabilizer) quantum error‐correc1ng codes.

2.  A protocol for extrac1ng the syndrome from each code.

3.  A (classical) decoding algorithm for interpre1ng the syndrome of each code.

4.  A finite universal gate basis and a protocol for implemen1ng each gate in encoded form.

Task: Simulate an ideal quantum circuit with faulty components. (E.g., inverse quantum Fourier transform) |k5〉 H • • • • •

|k4〉 π2 H • • • •

|k3〉 π4

π2 H • • •

|k2〉 π8

π4

π2 H • •

|k1〉 π16

π8

π4

π2 H •

|k0〉 π32

π16

π8

π4

π2 H

Applica:on of an FTQCP to an ideal quantum circuit: Step 1: Compile the ideal (g‐gate) circuit into a new (g’‐gate) circuit over the specified finite gate basis.

Step 2: Choose a code large enough to suppress errors below O(1/g’) with FTQEC.

Step 3: Replace each qubit and gate (incl. prep. & meas.) in the new circuit with encoded (“logical”) versions.

Step 4: Insert syndrome extrac1on aler each encoded gate. (Some may be able to be omimed.)

Step 5: Insert decoding‐iden1fied recovery before each non‐Clifford encoded gate. Step 6: Aler final logical measurement, perform final classical decoding of outcomes to infer result.

}}

FTQEC protocol

Encoded computa:on

Quantum compiling


•  Step 1: Compile the ideal circuit into a finite gate basis . The gate basis does not need to be physically realizable—we will simulate these “encoded” gates by physical gates, which may be different.

•  Note: This “quantum compiling” is itself a simula1on, which may generate approximaEon errors depending on the quantum circuit it simulates. If the ideal circuit has g gates, we want the approxima1on error to be ε = O(1/g).

•  Dawson‐Nielsen Solovay‐Kitaev quantum compiling algorithm: •  Finds ε‐approxima1on to of length in 1me .

•  Requires each gate to have its inverse in the finite gate basis.

Ugate O(log3.97(1/ε)) O(log2.71(1/ε))

|k5〉 H • • • • •

|k4〉 π2 H • • • •

|k3〉 π4

π2 H • • •

|k2〉 π8

π4

π2 H • •

|k1〉 π16

π8

π4

π2 H •

|k0〉 π32

π16

π8

π4

π2 H

G

Circuit to simulate now has gates. g′ = O(g log3.97 g)

•  Popular choices:

•  Real gate basis:

•  Kitaev gate basis:

•  Surface‐code cluster‐state gate basis:

•  Standard gate basis:

•  Overcomplete standard gate basis:

•  Well‐suited to Solovay‐Kitaev quantum compiling.

•  Magic‐state standard gate basis:

•  Very popular choice, but quantum compiling with this basis requires some finesse:

•  X, Z gates ignored: Classical update of “logical Pauli frame.”

•  S, T, and H gates by the following circuits, which rely on “magic states.”

Finite universal gate bases


GCSS︷︸︸︷{CNOT, |0〉, |+〉, MX , MZ}∪{| + i〉}︸︷︷︸

GClifford

∪{|T 〉}

{H,TOF } ∪ {|0〉, MZ}

{H,Λ(S)} ∪ {|0〉, MZ}

{I, X, Z,H, S, S†, T, T †,CNOT } ∪ {|0〉, |+〉, MX , MZ}

|θ〉 :=1√2(|0〉+ eiθ|1〉)

|+〉 := |2π〉

|T 〉 := |π/4〉

|+i〉 := |π/2〉

{H,Λ(Z)} ∪ {|+〉, MX , MZ} ∪{ M(X±Y )/√

2}

{H,T,CNOT } ∪ {|0〉, MZ}

Λ(U) :=[1 00 U

]S := Λ(i)

H = (X + Z)/√

2CNOT := Λ(X) TOF := Λ(CNOT )

T := Λ(eiπ/4)

|ψ〉 • S T |ψ〉

|π/4〉 !"#$%&'( MZ •

|ψ〉 • Z S |ψ〉

|π/2〉 !"#$%&'( MZ •

|ψ〉 S !"#$%&'( X S H |ψ〉

|+〉 • S MZ •

Nota1on legend:

Warning: not a Clifford circuit.

Infinite code family


Approach 1: Concatenate a finite (stabilizer) quantum error‐correc1ng code •  FTQCP possible for any stabilizer code as a base, but CSS codes are par1cularly nice.

•  Nomenclature: “Level ” of concatena1on is an ‐qubit block. (Level 0 is “physical.”)

•  Error‐detec:ng codes: lower levels locate errors for upper levels, so error‐detec1ng codes can correct them.

•  Numerous codes studied in the literature: •  [[2, 1, 1]] Bell‐state (for LOQC) •  [[5, 1, 3]] Five‐qubit

•  [[6, 2, 2]] C6 •  [[7, 1, 3]], [[49, 1, 9]] Steane/color (and concatenated Steane)

•  [[4, 1, 2]] (“C4”) [[9, 1, 3]], [[25, 1, 5]], [[49, 1, 7]], [[81, 1, 9]] Bacon‐Shor

•  [[15, 1, 3]] Reed‐Muller •  [[13, 1, 3]], [[41, 1, 5]], [[85, 1, 7]] Surface

•  [[21, 3, 5]], [[60, 4, 10]] Reed‐Simon (“polynomial”)

•  [[21, 3, 5]], [[23, 1, 7]] Golay

•  [[43, 5, 7]], [[45, 3, 9]], [[47, 1, 11]], [[75, 5, 11]], [[77, 3, 13]], [[79, 1, 15]], [[97, 7, 13]], [[99, 5, 15]], [[101, 3, 17]], [[103, 1, 19]] Quadra1c residue

•  [[31, 11, 5]], [[63, 27, 7]], [[63, 39, 5]], [[79, 1, 15]], [[89, 1, 17]], [[103, 1, 19]], [[127, 1, 19]], [[127, 29, 15]], [[127, 43, 13]] BCH

! n!

Infinite code family


Approach 2: Tesselate a finite (stabilizer) quantum error‐correc1ng code •  Goal is to have local checks and global logical operators.

•  Nomenclature: “Homological” (i.e., boundary) codes: logical ops are boundaryless, checks are boundaries.

•  Can exist in 1D, 2D, 3D, etc.; research focus is on 2D

•  Not all such codes are good for FTQC (e.g., Bacon‐Shor codes)

•  Finding new topological codes an ac1ve research area: •  4.4.4.4 Surface code •  3.4.6.4 Surface code

•  3.6.3.6 Surface code

•  4.8.8 Color code

•  6.6.6 Color code

•  3.12.12 Color code •  4.4.4.4 Bacon‐Shor code

•  3.4.6.4 Bombin topological subsystem code

•  Suchara‐Bravyi‐Terhal hypergraph topological subsystem code

•  3D Haah codes

•  3D, 4D, 5D, etc. surface codes and color codes

3D color code: [[15, 1, 3]] Reed‐Muller code. Figure from [Bombin & Mar1n‐Delgado, PRL 98, 160502 (2007)]

These 2D color codes extend the Steane code by tessela1on.

Performance of code families


Loosely speaking, if each [circuit opera1on, qubit] fails with probability p, the probability that the [fault‐tolerantly, ordinarily] error‐corrected qubit fails, denoted pf, qualita1vely looks as follows:

•  Finite‐size effects cause the intersec1on point to dril: “pseudothreshold.” •  Topological codes have stat. mech. analogues, enabling scaling Ansätze near threshold.

p

p fail

p

p fail

p

p fail

Concatenated codes Surface, color codes Bacon‐Shor codes

•  Curves steepen with code length. •  Mutual intersec1on at p = pf.

•  Curves steepen with code length. •  Mutual intersec1on at inflec1on.

•  Curves steepen with code length. •  Intersec1on with p = pf drils to 0.

Warning: Error threshold (code capacity) and accuracy threshold (fault‐tolerant) curves look similar. Be careful when reading/quo1ng literature.

A note on noise models


•  Local stochas:c noise

•  For any r specified loca1ons in the circuit, the sum of the probabili1es of all fault paths with faults at those r loca1ons is no larger than pr.

•  This noise model is preserved under concatena1on.

•  “Non‐Markovian”

•  Noise describable by a system‐bath Hamiltonian.

•  Leakage errors ignored in this tutorial; later talks at QEC will study them. •  Local iid depolarizing channel

•  Each prepara1on and gate followed by a depolarizing channel Pm(p).

•  Each measurement preceded by Pm(p) and result flipped with probability p.

•  Local bit‐flip‐plus‐phase‐flip channel

•  Each prepara1on and gate followed by bit‐flip channel B(p) followed by phase‐flip channel Φ(p).

•  Each measurement preceded by B(p) and Φ(p) and result flipped with probability p.

•  Phenomenological syndrome error model

•  En1re sub‐circuit for extrac1ng a syndrome bit causes no data errors but flips syndrome bit with probability p.

•  Not par1cularly realis1c when combined with the first two models, but enables mapping onto random‐interac1on sta1s1cal mechanics model for greater analysis.

Popular for numerical es:ma:ons of the accuracy threshold

A CSS decoder that separates bit‐flip and phase‐flip QEC treats the former noise model like the laXer, missing the correlaEons in Y = iZX errors. MulEply the former threshold by 2/3 to compare to laXer threshold for phenomenological noise.

Popular for provable lower bounds on the accuracy threshold

Syndrome extraction


General (fault intolerant) circuit for measuring a Pauli operator P:

|0〉 H • H !"!!!|ψ〉 P

Problem: Even perfect gates can cause errors to spread badly!

for factors into product of operators. P = P1 ⊗ · · ·⊗ PnΛ(P ) Λ(Pi)

|0〉 H • • X • • H !"!!!#$%&'()*#$%&'()*#$%&'()*#$%&'()* =

|0〉 H • • • X • H !"!!!#$%&'()*#$%&'()* X#$%&'()*#$%&'()*=

|0〉 H • • • • H Z !"!!!#$%&'()* X#$%&'()* X#$%&'()*#$%&'()* =

|0〉 H • • • • H !"!!!#$%&'()* X#$%&'()* X#$%&'()*#$%&'()*Possible solu:ons:

1.  Shor extrac:on: requires w‐cat state ancilla for weight‐w Pauli operator.

2.  Steane extrac:on: requires n‐qubit encoded or for n‐qubit code.

3.  Knill extrac:on: requires 2n‐qubit encoded for n‐qubit code.

4.  Topological extrac:on: requires extrac1on schedule to quench error spread on lauce.

|0〉 |+〉|Φ+〉 := 1√

2(|00〉 + |11〉)

Correlated error!

Concatenated-code extraction


Shor extrac:on: •  Works for all stabilizer

codes.

•  Need to repeat t + 1 1mes to boost reliability of outcome.

• H !"!!!˛̨e+

¸ • H !"!!!• H !"!!!

• H !"!!!#$%&'()* #$%&'()* #$%&'()* #$%&'()*

Steane extrac:on: •  Works for all CSS

codes

•  Z‐check circuit is similar:

• H !"!!!• H !"!!!

• H !"!!!|+̄〉 • H !"!!!

• H !"!!!• H !"!!!

• H !"!!!#$%&'()* #$%&'()* #$%&'()* #$%&'()* #$%&'()* #$%&'()* #$%&'()*

Knill extrac:on: •  Works for all CSS codes.

•  Teleports logical qubit: Could teleport onto an encoded gate and do QEC & gate simultaneously.

•  Measures both X and Z checks in parallel. |0̄0̄〉+|1̄1̄〉√

2

• H !"!!!• H !"!!!

• H !"!!!• H !"!!!

• H !"!!!• H !"!!!

• H !"!!!#$%&'()* !"!!!#$%&'()* !"!!!#$%&'()* !"!!!#$%&'()* !"!!!#$%&'()* !"!!!#$%&'()* !"!!!#$%&'()* !"!!!

|+̃〉 :=1√2(|0 . . . 0〉+ |1 . . . 1〉)

•  1 gate/data error propagates to at most 1 error.

•  Correlated ancilla errors could propagate badly: need to disEll them (more later on this).

•  Extrac1on done at each level independently, assuming the encoded gates are available

¯|0〉 • H !"!!!∣∣ψ̄⟩ #$%&'()*

Circuits for these X checks (detec1ng Z errors) from [Eas1n, PhD thesis (2007); arXiv:0710.2560]

Topological-code extraction


Topological extrac:on rules: •  Let errors propagate, if they do so gracefully.

•  No elaborate ancillas needed!

|0〉 !"#$%&'( !"#$%&'( !"#$%&'( !"#$%&'( )*+,MZ

••

••

Some figures adapted from figures by Jonas Anderson.

|+〉 • • • • !"#$MX%&'()*+,%&'()*+,%&'()*+,%&'()*+,Z4 check (for X errs) X4 check (for Z errs)

1.  Code checks propagate to

themselves

2.  Syndrome checks propagate to themselves mul1plied a stabilizer element

“Clock” schedule

“Book” schedule

1.  Code checks propagate to themselves

2.  Syndrome checks propagate to themselves mul1plied a stabilizer element

4.4.4.4 surface code, with qubits on ver1ces.

Topological-code extraction


•  Must repeat extrac1on (d 1mes suffices) to localize syndrome bit errors in 1me: Send changes in syndrome bits to decoder.

•  Concatenated codes using Steane or Knill extrac1on method do not require syndrome‐extrac1on repe11on, but they are resource‐intensive via post‐selec1on: they require verified ancilla states.

a.)

b.)

time

time

4.4.4.4 surface code. (a) syndrome differences (b) decoding‐inferred recovery. Figure from [Fowler et al., PRA 80, 052312 (2009)]

Ancilla verification


Concatenated‐code syndrome extrac:on works if ancilla has no correlated Z errors. How to achieve this? •  Method 1: Shor verificaEon: Verify each syndrome bit separately.

a.  If is a cat state, prepare in cascaded method and add a single CNOT gate to verify. Reject if fails verifica1on.

b.  Otherwise, perform Shor syndrome extrac1on on using verified cat states. Repeat each extrac1on t+1 1mes for a distance 2t+1 code.

|+〉

|+〉

|+〉

Figure from [Eas1n, PhD thesis (2007); arXiv:0710.2560].



Concatenated‐code syndrome extrac:on works if ancilla has no correlated Z errors. How to achieve this? •  Method 2: Steane verificaEon: Verify the syndrome‐at‐once.

a.  For distance 3 code, perform Steane Z syndrome extrac1on (with unverified ancillas) on two states, then use one to perform Steane X syndrome extrac1on on the other. Reject if fails verifica1on.

b.  For higher distance codes, addi1onal elaborate recursive checking is necessary

|+〉

Figure for Steane verifica1on of [[23, 1, 7]] Golay code from [Paetznick & Reichardt, arXiv:1106.2190].

|+〉

|+〉 • !"#$%&'( |+〉

|+〉 !"#$%&'( MZ

|+〉 • • MX

|+〉 !"#$%&'( MZ



Concatenated‐code syndrome extrac:on works if ancilla has no correlated Z errors. How to achieve this? •  Method 3: LaEn‐square (Steane) verificaEon: Force weight‐s errors lead to weight‐at‐least s syndromes.

•  Method 4: Ancilla decoding: Remove “verify” from “prepare – verify – couple – decode” chain.

•  Method 5: Overlap method: Exploit overlaps of stabilizer generators.

•  Method 6: …

|+〉

Time to disEll generally scales quadraEcally with block size: Can dominate overhead of FTQEC.

In [Isailovic et al, Proc. 35th ISCA, 177‐188 (2008)], ancilla pipeline es1mated to take up to 68% of circuit footprint!

Reminder: topological codes do not require ancilla for syndrome decoding!

Decoding algorithm


General tradeoff: low decoding complexity vs. low decoding failure probability •  Op:mal decoding: Finds recovery most likely to succeed given the syndrome.

•  Most‐likely error (MLE) decoding: Finds error most likely to have occurred given the syndrome.

•  Not op1mal for degenerate codes. (But premy close in prac1ce.)

•  NP‐hard in general, even though it is not op1mal. •  CSS codes: decoder expressible as an integer program; threshold maps

to order‐disorder phase transiEon in an associated random‐interac1on classical sta1s1cal mechanics model.

•  Minimum‐weight perfect matching (MWPM):

•  MLE decoder reduces to this for surface codes, complexity is O(d7.5) (Micali‐Vazirani).

•  Fowler et al. recently tailored MWPM to surface codes, parallelized complexity is O(1).

•  Level decoding: Decodes concatenated codes at each level independently & simultaneously: not globally op1mal, but one can do op1mal decoding by lookup table for the base code given the level’s noise model.

•  Belief‐propaga:on / Renormaliza:on‐group decoding: Guesses a recovery at fine‐grained scale, passes messages to coarser‐grained scale and iterates, converging on a solu1on. Works for concatenated & topological codes.

•  Parallelized complexity is O(log d)

BP level decoding. Figure from [Poulin, PRA 74, 052333 (2006)]

Encoded gates


Two general methods: Transversal or code deforma:on

Turaev‐Viro code deforma1on. Figure from [Koenig et al., Ann. Phys. 325, 2707‐2749 (2010)]

“Pancake” architecture of interleaved data blocks and syndrome blocks. Transversal opera1ons are ver1cal between layers. Without a 3D architecture, addi1onal SWAP opera1ons are needed.

Transversal gates


•  X, Z, CNOT: for any stabilizer code

•  “Destruc:ve” MX, MZ: for any stabilizer code

All computaEon is transversal except preparaEon of encoded magic states. To get high‐fidelity magic states, we encode them first and then disEll them (transversally) in encoded form.

••

••

••

•

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

•  X, Z: for odd‐length CSS codes.

•  H: for strong CSS codes.

•  S: for doubly‐even CSS codes of length (via ).

•  T: for quadruply‐even CSS codes of length (via ).

With encoded “magic states”: transversal nondestruc1ve MX, MZ

|ψ〉 • S T |ψ〉

|π/4〉 !"#$%&'( MZ •

|ψ〉 • Z S |ψ〉

|π/2〉 !"#$%&'( MZ •

|ψ〉 S !"#$%&'( X S H |ψ〉

|+〉 • S MZ •

(2! + 1) mod 4 Λ(i2!+1)

Λ(i(2!+1)/2)(2! + 1) mod 8

Eas:n‐Knill Theorem: Impossible to get transversal universal gate set. (3D color codes: can make QEC the only non‐transversal opera1on, though.)

}

Transversal gate: A gate that acts independently on each physical qubit in a code block. Strictly transversal gate: An encoded gate that acts idenEcally on each physical qubit in a code block.

Transversal Strictly transversal

For X, Z strictly transversal

With encoded “magic states”: transversal S, T, H

Not necessary to include, but convenient for concatenated codes.

|ψ〉 • MZ |ψ〉|+〉 !"#$%&'( MZ

|ψ〉 !"#$%&'( MX |ψ〉

|0〉 • MX

Code deformation


•  In a surface code, we can remove a pair of X or Z checks and create a logical qubit. •  By judicious measurements, we can effec1vely move the “defects” or “holes” around the surface, akin to

cluster‐state quantum computa1on.

•  Depending on the types of checks involved, braiding one defect around another can perform a CNOT gate.

•  Using magic states for all other gates, one can get universal encoded computa1on solely by this “code deforma1on.” (Turaev‐Viro non‐stabilizer codes can achieve universal QC by code deforma1on alone!)

•  More details in the next talk by Robert Raussendorf.

Figure from R. Raussendorf’s talk following this one. Ask me about my 1me machine in the Q&A session.

Magic-state injection


Injec:on by teleporta:on: •  Presumes an independent method of preparing encoded .

•  “Un‐encode” means coherent encoding circuit run in reverse.

•  Half of Bell pair is very exposed once un‐encoded.

|M〉 • MX!"#$%&'(

∣∣0⟩

/ H • )*+,Unencode -./01234 MZ •∣∣0

⟩/ -./01234 X Z

∣∣M⟩

! ! ! ! ! ! ! ! ! ! !"""""

"""""

! ! ! ! ! ! ! ! ! ! !

Injec:on by code deforma:on: •  First copy state to a single defect, independently prepared in

encoded .

•  Grow the defect, increasing its protec1on.

•  State is very exposed at early stages of growth.

|0〉

|0〉

Magic-state distillation


To dis:ll states, run Steane coherent encoding circuit in reverse:

To dis:ll states, run Steane coherent encoding circuit in reverse:

|π/4〉 |π/2〉

∣∣∣π̃/4⟩

• MX

∣∣∣π̃/4⟩

• MX∣∣∣π̃/4

⟩ !"#$%&'( !"#$%&'( • |π/4〉∣∣∣π̃/4

⟩• MX

∣∣∣π̃/4⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/4⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/4⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/4⟩

• MX

∣∣∣π̃/4⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/4⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/4⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/4⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/4⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/4⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/4⟩ !"#$%&'( !"#$%&'( !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/2⟩

• MX

∣∣∣π̃/2⟩

• MX∣∣∣π̃/2

⟩ !"#$%&'( !"#$%&'( • |π/2〉∣∣∣π̃/2

⟩• MX

∣∣∣π̃/2⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/2⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

∣∣∣π̃/2⟩ !"#$%&'( !"#$%&'( !"#$%&'( MZ

Dis:lla:on threshold for : . •  Can be made provably 1ght to

with a Steane unencoder.

|π/4〉

Ac1ve research: reducing the overhead cost.

(√

2− 1)/2√

2 ≈ 14.6%

Dis:lla:on threshold for : •  Can be made provably 1ght to 50% Repeat disEllaEon O(log g’)

Emes, for each T gate in compiled quantum circuit

Repeat disEllaEon O(log g’) Emes, for each S gate in compiled quantum circuit

If encoded H is available directly, can reuse state for each S gate in compiled quantum circuit:

|π/2〉 !"#$%&'( H !"#$%&'( H |π/2〉

|ψ〉 • • S |ψ〉

|π/2〉

≈ 14.1%

If any round of dis:lla:on fails, start all over!

Subtlety: Need to first pre‐twirl inputs by A, measure Z checks, apply A−1, then measure X checks, and only con1nue to these circuits if checks show no error. for states and for states.

A = e−iπ/4SXA = Y|π/4〉 |π/2〉

Accuracy thresholds


The Monte Carlo methods used in the literature to es:mate the accuracy threshold vary and are elaborate‐‐‐worthy of a tutorial unto themselves.

The analysis techniques used in the literature vary even more widely, using techniques like “Extended rectangles,” “Malignant pair coun:ng,” “Self‐avoiding walks.” Again, a subject worthy of a tutorial in its own right.

Es:mates/strict lower bounds of the accuracy threshold vary widely and are heavily dependent on model assump:ons. The highest reported values are in the 3% range while the lowest reported values are in the 10‐6 range.

Meaningful upper bounds on the accuracy threshold are par:cularly hard to come by. The lowest upper bound established to date (also model dependent, of course) is , if the only two‐qubit gate used is CNOT.

1− 1/√

2 ≈ 29.3%

Synopsis


Four components of every known fault‐tolerant quantum compu:ng protocol (FTQCP) 1.  An infinite family of (stabilizer) quantum error‐correc1ng codes.

2.  A protocol for extrac1ng the syndrome from each code.

3.  A (classical) decoding algorithm for interpre1ng the syndrome of each code.

4.  A finite universal gate basis and a protocol for implemen1ng each gate in encoded form.

Applica:on of an FTQCP to an ideal quantum circuit: Step 1: Compile the ideal (g‐gate) circuit into a new (g’‐gate) circuit over the specified finite gate basis.

Step 2: Choose a code large enough to suppress errors below O(1/g’) with FTQEC.

Step 3: Replace each qubit and gate (incl. prep. & meas.) in the new circuit with encoded (“logical”) versions.

Step 4: Insert syndrome extrac1on aler each encoded gate. (Some may be able to be omimed.)

Step 5: Insert decoding‐iden1fied recovery before each non‐Clifford encoded gate. Step 6: Aler final logical measurement, perform final classical decoding of outcomes to infer result.

Fault-tolerant classical computing

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