Quantum Error Correction B.M. Terhal, IQI, RWTH Aachen

I Formalism/Basics of (stabilizer) codes II QEC Conditions, Misc. Interesting Codes, Universal Encoded QC III QEC with surface code See also my review, arxiv.org: 1302.3428

What is a (Subspace) Code?

Specification of a subspace, say 2-dimensional, in a larger space. Basis in this 2-dim space given by encoded states |0 , |1 . Logical operators 𝑋 , 𝑍 with 𝑋 𝑍 = −𝑍 𝑋 . By definition, we have 𝑍 |0 = |0 , 𝑍 |1 = −|1 , 𝑋 |+ = |+ , 𝑋 |− = −|− , Specification? Simple for stabilizer codes (say, on n qubits)

Qubit stabilizer codes Pauli group P, group of products of Pauli operators and ± 𝒾I acting on n qubits, e.g. 𝐼1⨂𝑋2⊗𝑍3…⊗ 𝐼𝑛

Take Abelian subgroup S E.g. 𝑆 =< 𝑋1𝑋2, 𝑍1𝑍2 > (notation: <A,B,…> is the group generated by A,B,…) Codespace C is +1 eigenspace of group S (but other eigenspaces of equally good codes): 𝐶 = { 𝜑 |∀ 𝑠 ∈ 𝑆, 𝑠 𝜑 = |𝜑 } E.g. 𝐶 = |00 + |11 : 1-dimensional.

S

P

Parity Checks Generators of S can be viewed as (quantum) parity checks. Eigenvalues are always ±1. E.g. 𝑆 =< 𝑋1𝑋2, 𝑍1𝑍2 >.

Dimension of C? With k linearly-independent checks on n qubits, the subspace C is 2n-k-dimensional, encoding n-k qubits.

Non-stabilizer codes: checks still commute but not Paulis. Stabilizer codes on qudits and oscillators.

Logical Operators i=1..n-k, 𝑋 𝑖 , 𝑍 𝑖 . How to find them? In the Pauli group.… ∀𝑠 ∈ 𝑆, 𝑠 𝑋 0 = 𝑠(|1 ) = |1 .

• 𝑠𝑋 = 𝑋 𝑠: correct. Thus logical operators commute with all check operators.

• Logical operators are not (products of) check operators (otherwise trivial on codespace):

Operators in P which commute with all 𝑠 𝜖 𝑆: centralizer C(S)=<S, 𝑋1, 𝑍1…𝑋𝑛−𝑘 , 𝑍𝑛−𝑘> of S in P

• Logical operators non-unique:

𝑋 ≡ 𝑋 𝑠 for any 𝑠 ∈ 𝑆.

Errors and Protection Qubits such as superconducting/spin qubits

dephase (T2, Z errors) and decohere

(T1, Z and X errors) within less than 1 millisec.

Gates/measurements are realized approximately.

How is one to make a computer out of these fragile components?

Use many elementary qubits to encode one better-protected qubit.

Simple model: assume that noise/decoherence is independent & uncorrelated for each elementary qubit (or component). Not always true…

1

Pauli Errors (more general in Lec. II)

Elementary qubits only undergo bit flip X,

phase flip Z or both (Y).

Weight of a multi-qubit Pauli (error) E is |E|: # qubits on which it is nontrivial. (e.g. weight of 𝑋1𝑍3 is 2)

Error Model: Probability (error E) ∼ 𝐶𝑝|𝐸| with some constant C and 0 ≤ 𝑝 ≤ 1.

Choose codespace C (n qubits) such that low-weight (less than t) errors are either harmless (trivial) or can be detected and corrected. Probability (logical error) ∼𝐶′ 𝑛 𝑝𝑡+1 for some constant 𝐶′ 𝑛 .

Compare logical error rate with basic error rate 𝐶 𝑝.

1

Pauli Errors

Consider error E on encoded state: E|𝜑 . 1. Either E commutes with all 𝑠 𝜖 𝑆. Then E is trivial (in S) or a logical operator (bad!). 2. Or, ∃ 𝑠 ∈ 𝑆, 𝑠𝐸 = −𝐸𝑠, 𝑠 𝐸 𝜑 = −𝐸|𝜑 . We can detect the error. Distance of code is 𝑑 = min

𝑃 𝑃 , minimum weight of logical

error. Code can detect all errors of weight less than d. Correction? → Measure parity checks. Look at Shor’s [[9,1,3]] code [[n=# qubits, k=# encoded qubits, d=distance]].

Correction for Shor Code

Bacon-Shor code [[𝑛2, 1, n]]. Take n=3. 𝑆 =< ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑍𝑍𝑠, 𝑿=,1, 𝑿=,2 > with double row operators 𝑿=,1 = 𝑋1𝑋2𝑋3𝑋4𝑋5𝑋6, 𝑋4𝑋5𝑋6𝑋7𝑋8𝑋9

We have 𝑋 = 𝑋1𝑋2𝑋3, 𝑍 = 𝑍1𝑍4𝑍7. Distance is 3. 𝐸𝑟𝑒𝑎𝑙 = 𝑍1 happens…we measure 𝑿=,1 = −1,𝑿=,2 = +1. Conclude? We pick error 𝐸𝑔𝑢𝑒𝑠𝑠 (say 𝑍2) with 𝐸𝑟𝑒𝑎𝑙𝐸𝑔𝑢𝑒𝑠𝑠 𝜖 𝑆. GOOD

or we pick error 𝐸𝑔𝑢𝑒𝑠𝑠 (say 𝑍4𝑍7) with 𝐸𝑟𝑒𝑎𝑙𝐸𝑔𝑢𝑒𝑠𝑠 = 𝑃 . BAD

Decoding of error syndrome Minimum-weight decoding: Choose 𝐸𝑔𝑢𝑒𝑠𝑠 with min. weight.

Code with distance d=2t+1 can correct t errors.

Measuring Parity Checks

How to measure 𝑍1𝑍2 (parity) without measuring 𝑍1 or 𝑍2? General circuit for Pauli P which uses an ancilla qubit in |+ and collect parity info. in the phase of the ancilla. Circuit also projects the input state in an eigenstate of P=1 or P=-1 (with probabilities depending on error rate/amplitude). Discretization of continuum of errors onto Pauli errors!

Fast, high fidelity parity check measurements are THE crucial element in QEC.

Measurement is hard Imagined weak measurement trace with quantum jumps between -1 and 1, what happens at the jumps? How reliable

is error syndrome…

Experimentally attractive:

Couple ancilla to data and correct error without explicit measurement, i.e. entire evolution captured in

a Lindblad master equation (stabilizer pumping)

with e.g. 𝐿𝑗 ∝ 𝑃𝑐𝑜𝑟𝑟𝑒𝑐𝑡,𝑗(𝐼 − 𝑆𝑗). Small codes ok, but

larger (2D) codes require classical (non-local) logic before feedback.

Trouble with immediate feedback Multiple checks are measured and each separately corrected in closed-form:

Take [[9,1,3]].

𝑍1𝑍2 = −1, 𝑃𝑐𝑜𝑟𝑟𝑒𝑐𝑡 = 𝑋1 𝑍2𝑍3 = −1, 𝑃𝑐𝑜𝑟𝑟𝑒𝑐𝑡 = 𝑋3.

Not optimal!

Correct left or right qubit?

Wrong decision leads to logical error.

1D Ising Model Bacon-Shor code 𝑛2, 1, 𝑛 .

Horizontal ZZ-link checks on a row are like 1D Ising model terms on n spins.

-1 checks are domain walls.

Minimum weight error string

can only be found

by combining info on position

of domain walls (at least between neighboring checks)

Same issue with any stabilizer code.

3-qubit code with superconducting transmon qubits

Reed et al., Nature 482, 382-385 (2012)

Three-qubit bit flip code |0 = |000 , |1 = |111 . 𝑋 = 𝑋1𝑋2𝑋3, 𝑍 = 𝑍1.

Checks are 𝑍1𝑍2, 𝑍2𝑍3 (2-bits). When we decode, parity check info in 2 ancilla bits. Note 𝑃𝑐𝑜𝑟𝑟𝑒𝑐𝑡=Toffoli (CCNOT) gate (fidelity 78%) on decoded qubit Check that code can correct single qubit error.

3-qubit code with superconducting transmon qubits

Reed et al., Nature 482, 382-385 (2012)

Not at all what QEC will/should look like in the future:

• No encode/decode cycles as qubit is unprotected after decode.

• Issue with direct feedback needing non-local classical gates.

• We don’t need to correct as long as we know what errors have happened: Pauli frame…

Are these demonstrations really scalable?

II QEC Conditions, Miscellaneous Codes, Universality..

What is a subsystem (qubit stabilizer) code?

Non-Abelian group 𝐺 ⊆ 𝑃 with 𝐺 =< 𝐺1, 𝐺2…𝐺𝑚> where 𝐺𝑖 have, say, local support on a lattice.

Abelian center of group 𝑆 = 𝐺 ∩ 𝐶(𝐺).

Operators in 𝑃 which commute with elements in 𝑆: 𝐶(𝑆).

Operators in 𝐶 𝑆 \𝐺 (in 𝐶 𝑆 but not in G) are logical operators for protected qubits.

Operators in 𝐺\𝑆 are logical operators for gauge qubits (extra unused qubits).

Dimension of code space of n qubits?

k linearly independent generators of S

k+m linearly independent generators of G

Codespace dim=2𝑛−𝑘−𝑚

P C(S) G

P

S

Error Correction Noise Model: depolarizing noise with probability p, i.e.

On each qubit with probability (1-p) we have no error, with prob. p/3 X, p/3 Y and p/3 Z errors.

By measuring the eigenvalues of generators of S one can determine whether a state 𝜑′ is in the code-space or not. Error syndrome s.

With error syndrome s in hand, we decode, i.e. we determine the most likely error that could have occurred given syndrome s and we apply this on 𝜑′.

Decoding succeeds if the error 𝐸𝑟𝑒𝑎𝑙𝐸𝑔𝑢𝑒𝑠𝑠 ∈ 𝐺 (used to be S). Extra freedom in decoding!

Decoding fails when 𝐸𝑟𝑒𝑎𝑙𝐸𝑔𝑢𝑒𝑠𝑠 ∈ 𝐶 𝑆 ∖ 𝐺, i.e. a logical operator.

Measuring stabilizer operators Remember 𝑆 ⊆ 𝐺. Thus we can measure eigenvalues of generators of S by measuring local generators 𝐺𝑖 of 𝐺 =< 𝐺1, 𝐺2…𝐺𝑚>. Advantage!

But 𝐺𝑖 don’t commute (you can’t measure eigenvalues of X by measuring Y and Z), so one has to be careful..

[Necessary and sufficient condition for measuring

some 𝑠 = 𝐺𝑘 𝐺𝑘−1…𝐺1 by measuring 𝐺1 , …𝐺𝑘 sequentially is that [𝐺𝑗, 𝐺𝑗−1 …𝐺2 𝐺1]=0]

Bacon-Shor Code As a subsystem stabilizer code: [[n2,1,n]] Measure weight-2 horizontal ZZ and XX checks. Combine info to get eigenvalue of double row 𝑋=,𝑖 and 𝑍||,𝑗 checks

QEC Conditions Useful when we consider more general subspace codes, specific, non-Pauli, errors and approximate error correction. Assume superoperator description of noise:

𝑆 𝜌 = 𝐸𝑘𝜌

𝑘

𝐸𝑘†

i.e. depolarizing channel, independent X and Z errors. QEC Conditions. We encode bit strings 𝑖 → |𝑖 . One can correct the set of errors {𝐸𝑘} (or any linear combination of these errors) if and only if, for all k, l, we have

∀𝑖, 𝑗, 𝑖 𝐸𝑘†𝐸𝑙 𝑗 = 𝑐𝑘𝑙𝛿𝑖𝑗

QEC Conditions A. ∀𝑖, 𝑖 𝐸𝑘

†𝐸𝑙 𝑖 = 𝑐𝑘𝑙

where 𝑐𝑘𝑙 does not depend on i. If 𝑐𝑘𝑙 = 𝛿𝑘𝑙, errors send a codeword to orthogonal error-spaces (labeled by k), but errors may have identical effect on codewords.

B. 𝑖 𝐸𝑘†𝐸𝑙 𝑗 = 0,

orthogonal codewords are mapped by different (or the same) errors onto orthogonal states so that we can reverse the error (by a unitary transformation on codespace + ancilla). One could obey these conditions approximately, so that only approximate reversal is possible, leading to a reduction in error rate.

Beyond Pauli Errors Ex. 1: Two-level atom coupled to a zero temperature environment into which it releases energy. Dominant is error 𝜎− = |0 1|. Lindblad master equation:

𝜕𝜌

𝜕𝑡= −𝑖 𝐻, 𝜌 + ℒ 𝜌 with

ℒ 𝜌 = 𝐿𝑗𝜌𝑗 𝐿𝑗† −1

2𝐿𝑗†𝐿𝑗𝜌 −

1

2𝜌 𝐿𝑗†𝐿𝑗 , 𝐿𝑗 quantum jump operators

Ex. 1: Decay of 2-level atom: 𝐻 = −𝜔

2Z, 𝐿1 = 𝜅𝜎−

Ex. 2: Decay of bosonic mode (e.g. a cavity mode) into finite T reservoir

𝐻 = ω +1

2𝑎†𝑎, 𝐿1 = 𝜅− 𝑎, 𝐿2 = 𝜅+ 𝑎

†, 𝜅+ = 𝜅−𝑒−𝜔/𝑘𝑇

1

0

Beyond Pauli Errors Approximate protection against dominant errors? Integrate Lindblad master equation for short times 𝑡 = 𝜏

𝜕𝜌

𝜕𝑡= −𝑖 𝐻, 𝜌 + ℒ 𝜌

gives superoperator 𝑆𝜏 𝜌 = 𝐸𝑘𝜌𝑘 𝐸𝑘†

Ex. 1: Decay of 2-level atom: 𝐸0 = 𝜏𝜅𝜎−, 𝐸1 = 𝐼 −𝜏𝜅

2𝜎+𝜎−

Amplitude-damping Noise. 4-qubit code which brings error probability from 𝜏𝜅 down to (𝜏𝜅)2 per time-interval 𝜏. Ex. 2: Decay of bosonic mode in a cavity (H=0, rotating frame) at T=0:

𝐸0 = 𝜏𝜅− 𝑎, 𝐸1 = 𝐼 −𝜏𝜅−

2 𝑎†𝑎.

Qubit-into-oscillator code using cat states.

𝜋 − Cat-State Code Code A cat state is an equal superposition of 2 quasi-orthogonal coherent states.

The states |0 /1 + are even photon number states while |0 /1 − have an odd # photons. QEC conditions are met for 𝐸0 = 𝜏𝜅− 𝑎 (even ↔

odd, detect by measuring 𝑃 = 𝑒𝑖𝜋𝑎†𝑎) and 𝐸1 = 𝐼 −

𝜏𝜅−

2 𝑎†𝑎

(𝛼 → 𝛼𝑒−𝜏𝜅−/2) when 𝛼 → ∞.

Towards the 𝜋 − Cat-State Code

Paper from Nov. ’13: Measuring

𝑃 = 𝑒𝑖𝜋𝑎†𝑎 via coupling to

transmon qubit and readout of transmon qubit. An estimated 85% of the jumps are faithfully detected.

Qubit-into-Oscillator Code (Gottesman, Kitaev, Preskill 2000)

Momentum and position (quadratures)

𝑝 =𝑖

2𝑎† − 𝑎 , 𝑞 =

1

2𝑎† + 𝑎 , 𝑞, 𝑝 = 𝑖.

Code offers protection against any error which is a lin. combination

of small shifts 𝑒𝑖 𝑞 , 𝑒𝑖𝛿𝑝 (and products thereof) with 휀 , 𝛿 < 𝜋/2. E.g. • Equilibration of bosonic mode at T> 0 for small rates 𝜅±𝜏. Quantum jump operators (→ Kraus elements) can be expanded in terms of small shifts.

• Small fluctuations in phase of oscillator (𝑒𝑖𝜃𝑎†𝑎).

• Undesired small nonlinearities 𝑒𝑖 𝜆(𝑎†𝑎)2

Codewords are unphysical, approximations need to be made.

Qubit-into-Oscillator Code Stabilizer checks of code: 𝑆𝑝 = 𝑒

−2𝑖 𝑝 𝜋 𝑜𝑟 𝑒−2𝑖 𝑝 𝛼 , 𝑆𝑞 = 𝑒2𝑖 𝑞 𝜋

𝑜𝑟 𝑒2𝑖 𝑝𝜋

𝛼 . States with 𝑆𝑝 = 𝑆𝑞 = 1 have 𝑝, 𝑞 = 𝑘 𝜋, integer k.

We have 𝑍 = 𝑒𝑖 𝜋𝑞 , 𝑋 = 𝑒−𝑖 𝜋𝑝: 1 qubit is encoded. Error Correction: measure 𝑞 𝑚𝑜𝑑 𝜋 (𝑞 ∈ [0, 𝜋) ) and apply smallest shift to get back to 𝑞 = 𝑘 𝜋. Same for p. Shifts by less than 𝜋/2 are correctable.

Parameter 𝛼 = 𝜋. Codeword |0 is sum over 𝛿(𝑞 − 2𝑘 𝜋) in q-space (and 𝛿(𝑝 − 𝑘 𝜋) in p-space) Codeword |1 is sum over 𝛿(𝑞 − (2𝑘 + 1) 𝜋) in q-space (shifted) (and (−1)𝑘𝛿(𝑝 − 𝑘 𝜋) in p-space)

Qubit-into-Oscillator Code Codewords require preparing a superposition of perfect position eigenstates (infinite squeezing). Approximate realizations are Gaussian wavepackets in a Gaussian envelope. Creation of such states for cavity modes may be of future interest (e.g. using cat states, squeezing and homodyne detection)

|0

|1

q

Fault-Tolerance Given a [[n,k,d]] stabilizer code with d=2t+1 Assuming parity check measurement are perfect, logical error rate 𝑝 ~𝐶 𝑝𝑡+1. For t=1, one can use [[7,1,3]] or [[9,1,3]], but then what?

Code Concatenation: 𝑛1, 1, 𝑑1 , 𝑛2, 1, 𝑑2 → 𝑛1𝑛2, 1, 𝑑1𝑑2

with lower logical error rate. E.g. [[9,1,3]] concatenated with itself. Parity check 𝑍1𝑍2 → 𝑍 1𝑍 2, weight 6! Parity check measurements are not fault-free:

• Syndrome information is not correct → REPEAT • Worse: single errors can feed back to data and kill

code advantage. Complicated ancilla needed.. Low-weight parity check + high distance codes? Topological Codes

Code Overhead in Concatenation

Overhead in code concatenation (t=1): with r >1 levels of concatenation,

we get error rate ( 𝐶 𝑝)2𝑟= 휀 per fault-tolerant realization of a

logical gate of size 𝑆𝑟 where S is the size of the basic fault-tolerant gate (including EC) Fault-tolerant realization of a circuit of size N with overall

accuracy 𝛿 ≈ 𝑁휀 requires thus 2𝑟 =log𝑁

𝛿

log1

𝐶𝑝

or the size of the

fault-tolerant circuit scales as 𝑁2𝑟 log 𝑆 = 𝑐(𝛿, 𝐶, 𝑝)𝑁(log𝑁)log (𝑆).

• Practical problem for concatenation: S can be large.

• Topological codes have different overhead issues, see last lecture

Computation: Encoded Gates Transversal gates, e.g. a transversal CNOT

On the classical repetition code |0 = |00…0 , |1 = |11…1 classical computation can be done transversally.

Theorem: for stabilizer codes not all gates can be done transversally.

Complicated non-transversal gate constructions can depress the noise threshold. Two better ideas: gates by code deformation and gates by injection-and-distillation.

Why focus on stabilizer qmemory+ (Clifford Substrate)

For stabilizer codes, (some) Clifford gates (which map Paulis onto Paulis) can be done with minimal overhead, e.g. CNOT and Hadamard for surface code (next lecture).

For surface code, 𝑇 = 𝑒𝑖𝑍𝜋/8 and S = 𝑒𝑖𝑍𝜋/4 gates can be done using 3 ideas: gate teleportation, code injection & magic state distillation → universality.

A single T gate requires 100s of noisy T-ancillas which are distilled into 1 (compare: a CNOT does not require any qubit overhead).

Fast decoding for Clifford substrate is not necessary (never need to do any Pauli P due to Gottesman-Knill theorem, keep track of Pauli frame).

III 2D Topological Error Correction (Surface Code)

With pictures taken from various papers in particular Topological Quantum Memory by Dennis et al. (2001).

EC Logic

Other topological 2D codes: color codes (Bombin/Delgado

etc)

- Measure weight > 4-body

operators for EC

- Noise thresholds in

similar range as SurfCode

+ All Clifford gates (i.e. H,

CNOT, S) done topologically and transversally.

2D subsystem surface code (Bombin/Poulin/Bravyi/Suchara)

+ Measure weight-2 parity

checks for EC

+ Noise threshold pc0.6%

- Qubit overhead[[41,1,3]]

+ Clifford gates CNOT and

H done topologically

2D Levin-Wen models or (non-stabilizer) Turaev-Viro codes (Kuperberg/Reichardt/Koenig)

- Measure weight-12

checks for EC. EC? Better threshold than SurfCode?

+ Universal set of gates by

code deformation!

1D codes - Distance is O(1), bad.

Bacon-Shor Code + Measure weight-2 parity

checks

- Non-topological:

𝑝 𝑛 → 0, 𝑛 → ∞.

…

Surface Code

Example of [[13,1,3]] code, encodes 1 qubit into 13 qubits and has distance 3. The distance is the minimum weight of a logical operator.

Plaquette Operator Ap

Star Operator Bq

Features • Plaquette and star operators have eigenvalues ± 1.

• Plaquette and star operators all commute. • Plaquette and star operator generate an

Abelian group S, the stabilizer group. • The code space

𝐶 = 𝜑 : 𝐴𝑝 𝜑 = 𝜑 , 𝐵𝑞 𝜑 = 𝜑 , ∀𝑝, 𝑞 .

Codespace is +1 eigenspace of group S. • Dimension of code space for n qubits is 2𝑛−𝑘

where k is the # of lin. independent generators in S.

• For surface code n-k=1, for toric code n-k=2

Logical Operators

Logical operators commute with all elements in the stabilizer group S but are not in S…

Plaquette Operator Ap

Star Operator Bq

(Left) trivial loops made from plaquette operators.

(Right) non-trivial loop which cannot be made from plaquette operators, i.e. this operator is not in S, but commutes with S.

For 𝜑 in codespace, 𝐴𝑝 𝜑= 𝜑 and 𝐵𝑞 𝜑= 𝜑 so action of plaquette and star operator is trivial on codespace. Thus

𝑋 𝐵𝑞 𝜑= 𝑋 𝜑 and we can deform a logical operator by multiplying with 𝐴𝑝 and 𝐵𝑞.

Surface Code

Rough boundar Smooth

boundary

Logical operators for the surface code; they can be deformed but cannot be made of shorter length (i.e. distance of the code is L). General L x L lattice with L2+(L-1)2 links (qubits) and L(L-1) plaquette and L(L-1) star operators (all independent).

Rough boundary

Errors • Imagine only Pauli errors occur (say depolarizing noise model)… • Pauli errors by definition anti-commute with at least one element of the stabilizer group. Surface code: X errors anti-commute with some plaquette operators and Z-errors anti-commute with some star operators. If error E anti-commutes with some 𝑠 ∈ 𝑆, then 𝑠(𝐸 𝜑 = −𝐸 𝜑 ) or codestate with error has -1 eigenvalue with respect to s.

By measuring generators of S (plaquette & star operators), we get information about what errors occurred Error Correction.

Encoding of |0 (and |+ )

|0 is the unique +1 eigenstate of 𝑍 and all plaquette and star operators. |0 Preparation Procedure: Set all qubits to 0…00. Note that |0. . 00 is a +1 eigenstate of all (Z) plaquette operators and 𝑍 . So measure all star (X) operators which will have random eigenvalues ± 1 and correct these `errors’ by doing Pauli Zs. Similar for |+

( |+ = (|0 + 1 / 2 ). Measurement of 𝑍 and 𝑋 is the reverse of these operation (includes EC)

Quantum Memory 1. First we encode bare qubits into a sheet (prepare |0 , |1 , |− or |+ ).

2. Then we continuously do parity check measurements and collect syndrome data (no correction) for some time t.

3. We do 𝑀𝑍 or 𝑀𝑋 (i.e. measure individual qubits in X and Z basis).

4. We process at leisure the syndrome data record and determine whether a |0 was decoded as a |0 etc.

1/3/2014

Check Measurement Circuits for measuring plaquette (left) and star

operators (right)

|0

|0

Uses only CNOT and Hadamard gates! Measure all plaquette and star operators in 2 rounds..

1/3/2014

measure +/- ;

prepare |+>

measure +/- ;

prepare |+>

measure +/- ;

prepare |+>

measure +/- ;

prepare |+>

measure +/- ;

prepare |+>

measure +/- ;

prepare |+>

measure +/- ;

prepare |+>

measure +/- ;

prepare |+>

measure +/- ;

prepare |+>

measure +/- ;

prepare |+>

measure +/- ;

prepare |+>

measure +/- ;

prepare |+>

5x7

Measure

“plaquette operator”

Z1Z2 Z3Z4

Measurement of plaquette operators using local ancillas in 2D plane.

Black dots are data

qubits of code, open

dots are ancilla

qubits. Dashed open

dots are ancillas in

preparation for star

measurements.

1/3/2014

measure 0/1 ;

prepare |0>

measure 0/1 ;

prepare |0>

measure 0/1 ;

prepare |0>

measure 0/1 ;

prepare |0>

measure 0/1 ;

prepare |0>

measure 0/1 ;

prepare |0>

Measure

“star operator”

X1X2 X3X4

Measurement of star operators using local ancillas in 2D plane.

Black dots are ancillas

for star measurement,

Open dots are data

qubits and dashed

dots are ancillas in

prep./meas. for

plaquette operators.

Put ancillas on the plaquettes and vertices of the original lattice.

Decoding (For Toric Code) How to decode by minimum weight decoding in case of noisefree syndrome measurement and in case of noisy syndrome measurement

Measurement of star operators gives info about Z errors. Measurement of plaquette operators gives info about X errors. That is: we have two identical decoding problems which we solve separately.

Example of good decoding

Black Z-error string is succesfully decoded as blue Z-error string. Gray X-error string is succesfully decoded as red X-error string (on dual lattice).

Places where syndromes are -1 are sometimes called defects. Good decoding: annihilate defects trivially.

Failed decoding

Black Z-error string is unsuccesfully decoded as green Z-error string since the product of the two is a logical operator, a nontrivial loop around the torus. Failed decoding: pair of defects is annihilated via a non-trivial path.

Minimum Weight Decoding Choose minimum weight decoder,

i.e. determine error with minimum

weight |E| which is consistent with

syndrome. Min. weight decoding can correct

any error of weight less than distance/2 (=L/2 here).

A ‘gas’ of random errors (paired defects) with constant density below some threshold c can still be corrected.

The code can correct typical errors of weight L2, that is, much beyond the distance L.

One thus expects a phase transition at a certain noise rate pc

p < pc : most errors are successfully decoded and corrected.

p ≥ pc : most errors are not succesfully corrected.

Min. Weight Decoding for Toric Code Take toric code (surface code

is more involved but similar).

Even number of defects.

Define a (taxicab) distance d(i,j)

between all pairs (i,j) of defects (on a torus one has to take the minimum over 4 distances).

Then find a matching which minimizes

the sum over the distances between the matched pairs.

For n defects, Edmond’s min. weight perfect matching algorithm runs in time O(n3), thus for L x L lattice, O(L6).

For quantum memory, time is not a problem.

Faster renormalization group decoders do exist.

Noise Threshold for Toric Code

Assume we measure syndrome

by noisefree circuits.

Assume independent X

and Z errors (probability

p of Z error on each qubit and

prob. p of X errors in one time-step)

Decode after every step (instantly).

Critical or threshold noise rate pc is 11% using minimum weight decoding. Very high!

Noisy Error Correction But errors can occur in CNOT

gates, H gate, measurement

and preparation. Two effects:

• Error on ancilla can

propagate to 4 data qubits…

not too bad..

• Erroneous CNOTs, measurement etc. can lead to faulty error syndrome.

Solution: repeat syndrome measurement L times and decode using L measurements….How does this work?

Decoding in 3D

Measurement of plaquette operators. At red dots the syndrome value changes Errors that could have caused such syndrome. Vertical errors are syndrome measurement errors. So again we need to find a minimum-weighted (error) chain which connects the red defects.

1D cross-section… Syndrome measurements in

time. Qubits are on horizontal links.

Syndrome measurements are

on vertical links.

Grey lines indicate -1 syndrome.

Black lines are a possible error

consistent with syndrome.

Examples of red dots where syndrome

changes.

To decode, we need to trivially bring

pairs of defects together but now in

space-time (3D).

Surface Code with Noisy EC

The memory noise threshold (no gates) is estimated as being between 7.5 x 10-3 and 1.1 x 10-2. (depending on decoding cleverness). Assume syndrome failure rate=effective Z error rate.

Logical error rate 𝑝 ∽ exp −𝜅 𝑝 𝐿 , 𝜅 𝑝 = 0.8 𝑎𝑡 𝑝 = 𝑝𝑐/3. Surface Code: At L=6, for a depolarizing probability equal to 2 x 10-4

(applied at all gates in the parity check circuit), we can have a logical 𝑋 error rate of 10-7.

Coding Scheme I We can encode our data qubits

into smooth qubits of sufficient

large area and sufficiently far

apart (so distance is suff. high).

Rough qubits will be only used to

help do CNOTs on smooth qubits.

Note that we can move encoded smooth qubits around by doing SWAP gates which by themselves are done by

3 local CNOT gates.

SWAP

A simple sheet

Assume planar L x L lattice of qubits on links, smooth boundary everywhere. At corners, star operators are of weight 2. At edges, star operators are weight 3. Product of all star operators is I. We thus have L2 plaquettes, (L+1)2 stars, and 2 L(L+1) qubits and 1 linear dependency between the stars. So L2+(L+1)2-1-2L(L+1)=0 encoded qubits.

How to encode more qubits in surface code

What happens when we remove one plaquette say Ap* from stabilizer S? This plaquette operator becomes the logical operator 𝑍1 of a new encoded qubit. Find 𝑋1… . . 𝑋1 is orange X-string which ends at hole. New qubit has distance 4, bad…make big hole.

How to encode more qubits

Make a bigger hole by taking out a volume of plaquette (and star operators) and their qubits. Hole has smooth boundary: smooth hole. Logical operators are green 𝑍1 and orange 𝑋1. We won’t use first logical qubit (𝑍0 and 𝑋0) associated with entire surface so we can always freely multiply with these operators. Distance of new qubit: minimum of circumference of hole and distance to smooth boundary of hole.

𝑋0 𝑍0

Rough hole

Similarly, one can make a rough hole by removing a cluster of star operators. It will have a rough boundary. E.g. in figure: we remove 9 star operators, 4 plaquette operators and 12 qubits. Logical operators are green 𝑍1 to rough boundary (rough-to-rough) and orange 𝑋1 which encircles the hole. Distance is again minimum of distance to boundary or circumference of hole

More holes, new encoding

More holes, more qubits and we can deform their logical operators. What is 𝑋1𝑋2… Deform to the orange logical operator 𝑋𝑠𝑚𝑜𝑜𝑡ℎ which connects the two holes. No more reference to the boundary (At least when 2 x distance to boundary is more than distance between holes). Smooth qubit is encoding of two hole qubits, i.e. |0 ≡ |0 |0 and |1 ≡ 1 1 . 𝑍𝑠𝑚𝑜𝑜𝑡ℎ is either 𝑍1 or 𝑍2. Many smooth qubits can be encoded.

Rough Qubits

Similarly, with two rough holes, one can encoded a rough qubit. Deform green 𝑍1𝑍2 to green 𝑍𝑟𝑜𝑢𝑔ℎ

and 𝑋𝑟𝑜𝑢𝑔ℎ is any of 𝑋1 or 𝑋2.

Thus the encoding of such a rough qubit is |0 ≡ 0 0 + 1 1 and |1 ≡ 0 1 + 1 0 . Creating and moving holes = changing the code = changing which stabilizer generators we measure.

CNOT CNOT’s action on Pauli’s 𝑈𝑃𝑈† = 𝑃′

𝑋1⨂𝐼2 → 𝑋1⨂𝑋2 𝐼1⨂𝑋2 → 𝐼1⨂𝑋2 𝑍1⨂𝐼2 → 𝑍1⨂𝐼2 𝐼1⨂𝑍2 → 𝑍1⨂𝑍2

1

2

X-strings are orange. Z-strings are green. Control qubit is smooth qubit. Target qubit is rough qubit. We move a smooth hole around a rough hole. Deformation of stabilizer group S=S(t=0) S(t)S(t=T)=S.

𝑍1 𝑋1

𝑋2 𝑍2

Understanding CNOT: Rules

• Moving a (smooth) hole. String 𝑋1 is pulled along. • You can’t push a green hole ( 𝑍1 ) through a green string ( 𝑍2 ). That is: the dotted green string with the same endpoints as 𝑍2 will be 𝑍1𝑍2.

𝑍2

𝑍1

CNOT between smooth qubits

We can only do a CNOT between a smooth qubit as control and rough qubit as target qubit. Looks quite limited….(such gates all commute!) Needs a circuit trick… In topological representation of gates we have black world lines as the smooth qubit and blue world lines as the rough qubit.

smooth

rough

Teleportation Circuit Conversion

1. Circuit for converting a rough qubit to a smooth qubit. The smooth qubit can be prepared in the logical |+ state.

2. Circuit for converting a smooth qubit into a rough qubit. How to do CNOT between (say) 2 smooth qubits: Convert (target) smooth qubit into rough qubit using circuit 2, do CNOT and convert rough qubit to smooth qubit using circuit 1.

smooth

rough

smooth

rough

Coding Scheme II (Lattice Surgery)

• Double hole encoding of qubit inefficient in terms of overhead.

• Different idea: encode each qubit in a sheet (Hadamard gate ok).

• CNOT gate via merging and splitting of sheets using an ancilla qubit sheet.

• Use this circuit identity (modulo Pauli corrections on control and target qubit). MXX is measurement of 𝑃 = 𝑋⊗ 𝑋 etc.

Horsman et al., New Jour. Phys. 14, 123011 ,2012

Lattice Surgery Method

• How to do 𝑃 = 𝑋 ⊗𝑋 measurement on two sheets?

• Prepare pink qubits in between sheets in |0>.

• Do a rough merge between sheets: new Z-checks at boundary have +1 eigenvalue, new 4 X-checks at boundary have random eigenvalues but their product equals 𝑋 ⊗𝑋 of the 2 sheets.

• Now we got 1 sheet (1 qubit) and the value 𝑋⊗ 𝑋 = ±1.

• Split the sheets by decoupling pink qubits, i.e. measure them in Z-basis and stop measuring X-checks at boundary. What happens with values for 𝑋 ⊗𝑋?

• All measurements have to be repeated L times for robustness of Pauli frame.

Horsman et al., New Jour. Phys. 14, 123011 ,2012

𝑿

CNOT Horsman et al., New Jour. Phys. 14, 123011 ,2012

C

T

INT

Hadamard Gate

Step 1: remove qubit code patch from rest of lattice.

Step 2: Do Hadamards on code patch. The state 𝐻⊗𝑛|𝜑 is a codestate of a new code with stabilizers 𝐻𝐴𝑝𝐻,𝐻𝐵𝑞𝐻 , i.e. Z-

plaquette operator becomes X-star operator on dual lattice.

Step 3: Change new code back to old code, i.e. change rough vertical boundaries back to smooth vertical boundaries. Effect is that lattice is shifted in position.

Code Overhead for (Quantum) Memory Classical (linear binary) LDPC codes (n,k,d) have 1. parity checks which act on O(1) qubits (and each qubit in O(1)

parity checks)

2. Rate 𝑘

𝑛≥ 𝑐1, and

3. 𝑑

𝑛≥ 𝑐2.

4. Computationally-efficient decoders (linear in n)

Repeat parity checks for fault-tolerance. For sufficiently large, but constant n, 𝑝 ~𝑒−𝛼 𝑛 as we can correct a constant density of errors due to 3. and due to 1. faulty-parity checks map single errors onto weight O(1) errors. Constant Overhead (see Gottesman, arxiv.org:1310.2984)

Surface Code Overhead

Compare encoding k qubits into k L x L patches of surface code: 𝑝 ~𝑘 𝑒−𝛼′ 𝐿, implying that L should scale as ln 𝑘 , or spatial overhead for k qubits is 𝑘 (ln (𝑘))2. Not a constant overhead For codes defined on two-dimensional lattices, surface code is generic: Theorem (Bravyi-Poulin-Terhal): 𝑘𝑑2 = 𝑂 𝑛 . Recent work on quantum LDPC codes in a 4-dimensional hyperbolic space with good rate, distance scaling as 𝑛𝛼 (𝛼 < 1, but finite threshold) and efficient decoders (Hastings, arxiv.org: 1312.2546). What are the rate, overhead & parity checks concretely?

Why scalable QC will be hard to realize or why some theorists should be thinking about real quantum/classical

systems instead of imagined ones • The overhead/size problem. Surface Code: 100 physical qubits

per logical qubit. Superconducting Qubits: cavity is O(1) mm in size.

• The accuracy threshold problem. Fidelity per CNOT should be safely below 1% (e.g. 10-2 currently in SC qubits). We haven’t explored the high fidelity regime of quantum information where many approximations commonly done are inaccurate.

• The control line problem. Single qubit gates are done by external EM radiation (TM with microwaves in SC, TM with on/off DC for electron spin-qubits, laser/microwave in ion-trap qubits) so require single-qubit control lines. On-chip low-T signal generators? Qubits need to be calibrated requiring individual control.

• The ancilla distillation factory overhead problem for universal QC