an introduction to quantum error correction · quantum information processing two perspectives...
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Quantum information processing
Two perspectives
Physics: Realization of devices, quantum logical gates,quantum memory, · · · .Computer science: algorithmes and quantumcryptography.
State of the art in physics
Quantum communication over distances < 100km;Logical gates between a few qubits (less than 10);Still far from the requirements in universal computation:many thousands of qubits.
Main obstacle: decoherence!
State of the art in physics
M. Devoret and R. Schoelkopf, Science, Vol. 339 no. 6124, 2013.
1 Classical and quantum noise, decoherence
2 Basics of quantum error correction
3 Recent experimental development: major obstacles
4 Some new directions
Classical bit, classical noise
Classical bit: strongly dissipative bistable system
1 0
∆U
U (x)
x
kTbruit
mx + ηx + ∂∂xU (x) = 0
1
η: friction coefficient
Classical bit in the state 0 or 1:
1 Strong dissipation;
2 kBTnoise � ∆U;
Classical noise, classical error correction
Classical noise: bit-flip errors
0
1
0
1
1-p
1-p
p
p
Classical error correction
0 −→ 000 and 1 −→ 111;Majorty vote for correction;New error probability: p −→ 3p2 − 2p3 (probability of twoor more errors).
Quantum bit
(− ~2
2m4+ U(x)
)ψk (x) = Ekψk (x), |0〉 = ψ0(x), |1〉 = ψ1(x).
|0⟩|1⟩
ω01=(Ε1−Ε0)/ħ
U(x)
Quantum state: c0 |0〉+ c1 |1〉.
Quantum noise: open quantum systems1
1 Schrödinger: wave funct. |ψ〉 ∈ H or density op. ρ ∼ |ψ〉 〈ψ|ddt |ψ〉 = − i
~H |ψ〉 , ddt ρ = − i
~ [H, ρ], H = H0 + uH1
2 Entanglement and tensor product for composite systems (S,M):
Hilbert space H = HS ⊗HMHamiltonian H = HS ⊗ IM + Hint + IS ⊗ HM
3 Randomness and irreversibility induced by the measurement ofobservable O with spectral decomp.
∑µ λµPµ:
measurement outcome µ with proba.Pµ = 〈ψ|Pµ |ψ〉 = Tr (ρPµ) depending on |ψ〉, ρ just beforethe measurementmeasurement back-action if outcome µ = y :
|ψ〉 7→ |ψ〉+ =Py |ψ〉√〈ψ|Py |ψ〉
, ρ 7→ ρ+ =PyρPy
Tr (ρPy )
1S. Haroche, J.M. Raimond: Exploring the Quantum: Atoms, Cavities andPhotons. Oxford University Press, 2006.
Quantum noise: interaction with environment
State space: {ρ : H 7→ H | ρ = ρ†, ρ ≥ 0,Tr (ρ) = 1}, whereH = HS ⊗Henv.
Dynamics: Uτ = exp(−iτH/~), H = HS ⊗ Ienv + Hint + IS ⊗ Henv.
Error channel: E(ρS) = trenv[Uτ (ρS ⊗ ρenv)U†
τ
].
Error channel: operator-sum representation
Considering a basis {ek} of Henv:
E(ρ) =∑
k
〈ek |Uτ (ρ⊗ |e0〉 〈e0|)U†τ |ek 〉
=∑
k
EkρE†k ,
where Ek = (IS ⊗ 〈ek |)Uτ (IS ⊗ |e0〉). As a consequence to theunitarity of Uτ , we have
∑k E†
k Ek = IS.
Quantum noise: interaction with environment
Discrete dynamics
E(ρ) =∑
k
EkρE†k , Ek = (IS ⊗ 〈ek |)Uτ (IS ⊗ |e0〉),
∑k
E†k Ek = IS.
Continuous dynamics (τ → 0): Lindblad equation
ddtρ =
∑k
(LkρL†k −
12
L†k Lkρ−
12ρL†
k Lk ).
Correspondance with discrete case: E0 = I− dt2
∑k L†
k Lk ,Ek =
√dtLk , k = 1,2, · · · .
Example: energy decay of a qubit
State space: {ρ ∈ span{|0〉 , |1〉} | ρ = ρ†, ρ ≥ 0,Tr (ρ) = 1}.
Energy decay channel
ρk+1 = E(ρk ) = E0ρk E†0 + E1ρk E†
1,
where
E0 = |0〉 〈0|+√
1− p |1〉 〈1| =
(1 00√
1− p
),
E1 =√
p |0〉 〈1| =
(0√
p0 0
).
Dissipation towards ground state:
Pk+11 = 〈1| ρk+1 |1〉 = (1− p) 〈1| ρk |1〉 = (1− p)k+1 〈1| ρ0 |1〉
Pk+10 = 〈0| ρk+1 |0〉 = 1− Pk+1
1 = 1− (1− p)k+1 〈1| ρ0 |1〉 .
Example: bit-flip and phase-flip of a qubit
Bit-flip channel
ρk+1 = E(ρk ) = E0ρk E†0 + E1ρk E†
1,
E0 =√
1− p I =√
1− p(|0〉 〈0|+ |1〉 〈1|) =
(√1− p 00
√1− p
),
E1 =√
pσx =√
p(|0〉 〈1|+ |1〉 〈0|) =
(0√
p√p 0
).
Phase decay channel
ρk+1 = E(ρk ) = E0ρk E†0 + E1ρk E†
1,
E0 =√
1− p I =√
1− p(|0〉 〈0|+ |1〉 〈1|) =
(√1− p 00
√1− p
),
E1 =√
pσz =√
p(|0〉 〈0| − |1〉 〈1|) =
(√p 0
0 −√p
).
1 Classical and quantum noise, decoherence
2 Basics of quantum error correction
3 Recent experimental development: major obstacles
4 Some new directions
Quantum error correction (QEC): bit-flip channel
Quantum error correction
Protect any superposition state c0 |0〉+ c1 |1〉 without any knowledge of c0
and c1.
Idea
Inspired by classical error correction, encode c0 |0〉+ c1 |1〉 as
c0 |000〉+ c1 |111〉 = c0 |0〉 ⊗ |0〉 ⊗ |0〉+ c1 |1〉 ⊗ |1〉 ⊗ |1〉 .
Measuring a qubit: asking a qubit if it is in |0〉 or |1〉 erases theinformation by projecting the superposition on |000〉 or |111〉 (nomajority vote).
Parity measurements: asking instead if qubits 1 and 2 are in the samestates (parity of qubits 1 and 2: σz ⊗ σz ⊗ I) and the same question forqubits 1 and 3 (σz ⊗ I⊗ σz ).
Correction: if m0 = m1 = 1 then no correction; if m0 = −1,m1 = −1then flip qubit 1; if m0 = −1,m1 = 1 then flip qubit 2; if m0 = 1,m1 = −1then flip qubit 3.
QEC in practice
Quantum gates:
t c
C-NOTgate
UC-NOT c ⊗ t = c ⊗ c⊕ t
X ,Y ,Z , H gates
U =σ x ,σ y ,σ z ,
12
(σ x +σ z )
σ x =
0 11 0
⎛
⎝⎜⎞
⎠⎟,σ y =
0 −ii 0
⎛
⎝⎜⎞
⎠⎟,σ z =
1 00 −1
⎛
⎝⎜⎞
⎠⎟
QEC steps: |ψ〉 = c0 |0〉+ c1 |1〉
Ini$alize
ψ
QEC in practice
Quantum gates:
t c
C-NOTgate
UC-NOT c ⊗ t = c ⊗ c⊕ t
X ,Y ,Z , H gates
U =σ x ,σ y ,σ z ,
12
(σ x +σ z )
σ x =
0 11 0
⎛
⎝⎜⎞
⎠⎟,σ y =
0 −ii 0
⎛
⎝⎜⎞
⎠⎟,σ z =
1 00 −1
⎛
⎝⎜⎞
⎠⎟
QEC steps: |ψ〉 = c0 |0〉+ c1 |1〉
Encode
00
Ini$alize
ψ
QEC in practice
Quantum gates:
t c
C-NOTgate
UC-NOT c ⊗ t = c ⊗ c⊕ t
X ,Y ,Z , H gates
U =σ x ,σ y ,σ z ,
12
(σ x +σ z )
σ x =
0 11 0
⎛
⎝⎜⎞
⎠⎟,σ y =
0 −ii 0
⎛
⎝⎜⎞
⎠⎟,σ z =
1 00 −1
⎛
⎝⎜⎞
⎠⎟
QEC steps: |ψ〉 = c0 |0〉+ c1 |1〉
Encode
00
Ini$alize
ψMeasureSyndrome
00
𝑋
QEC in practice
Quantum gates:
t c
C-NOTgate
UC-NOT c ⊗ t = c ⊗ c⊕ t
X ,Y ,Z , H gates
U =σ x ,σ y ,σ z ,
12
(σ x +σ z )
σ x =
0 11 0
⎛
⎝⎜⎞
⎠⎟,σ y =
0 −ii 0
⎛
⎝⎜⎞
⎠⎟,σ z =
1 00 −1
⎛
⎝⎜⎞
⎠⎟
QEC steps: |ψ〉 = c0 |0〉+ c1 |1〉
Encode
00
𝑋 𝑋 𝑋
Correct&DecodeIni$alize
ψ ψ00
MeasureSyndrome
00
𝑋
ancillasprovideparityinforma2on
QEC in practice
Failure modes?
QEC beyond bit-flip errors: phase-flip errors
Phase-flip vs bit-flip
Bit-flip error channel: spanned by {√
1− p I,√
pσx}
Ebit-flip(ρ) = (1− p)ρ+ pσxρσx .
Phase-flip error channel: spanned by {√
1− p I,√
pσz}
Ephase-flip(ρ) = (1− p)ρ+ pσzρσz .
Phase-flip QEC
Similarly to bit-flip case, encode c0 |0〉+ c1 |1〉 as
c0 |− − −〉+ c1 |+++〉 , where |±〉 = 1√2(|0〉 ± |1〉).
Parity measurements: We measure σx ⊗ σx ⊗ I and σx ⊗ I⊗ σx insteadof σz ⊗ σz ⊗ I and σz ⊗ I⊗ σz for the bit-flip case.
QEC beyond bit-flip errors
Theory of QEC
Similarly to an error channel, the error correction (measurement andfeedback) can be modeled by a quantum operation:
ρ 7→ R(ρ) =∑
k
RkρR†k .
This corrects an error channel ρ 7→ E(ρ) if for any ρ in the code space
R ◦ E(ρ) = ρ.
Theorem: discretization of error channels
Assume that R is the error-correction operation for the error channel Emodeled by operators {Ek}. Suppose F to be another error channel withoperation elements Fj which are linear combinations (with complexcoefficients) of operators Ek . Then the error-correction operation R alsocorrects the error channel F .
Corollary: case of qubit
It suffices to correct the operations {I, σx , σz , σy = iσxσz}: any matrix E on C2
is a linear combination of these operators.
QEC beyond bit-flip errors: a full QEC codeFull Steane Code – Arbitrary Errors
Singleroundoferrorcorrec;on
1 Classical and quantum noise, decoherence
2 Basics of quantum error correction
3 Recent experimental development: major obstacles
4 Some new directions
Google/UCSB: bit-flip error detection
Only protects classical states |000〉 or |111〉 (no superpositions).
Offline correction (no feedback).
J. Kelly et al., Nature 519, 66-69, 2015.
IBM: bit-flip/phase-flip detection
2
Figure 1. Surface code implementation and error detection quantum circuit. a, Cartoon schematic of SC consisting ofalternating square tiles of X- (yellow) and Z- (green) plaquettes for detecting phase-flip (Z) and bit-flip (X) errors, respectively.Semi-circular pieces reflect parity checks at the boundaries of the lattice. These plaquette tiles can be mapped onto a lattice ofphysical superconducting qubits with appropriate nearest-neighbour interconnectivity, as shown in the layer labeled MAP. Here,there are code qubits (purple spheres), X-syndrome qubits (yellow) for phase parity detection of surrounding code qubits, andZ-syndrome qubits (green) for bit parity detection of surrounding code qubits. The physical connectivity for superconductingqubits can be realised via coupling every qubit to two quantum bus resonators, shown as wavy blue diamonds in the MAP. Thedevice studied in this work (false-colored optical micrograph in b) embodies two half-plaquettes of the SC as circled in a, andallows for independent and simultaneous detection of X and Z errors on two code qubits, shaded purple in b and labeled Q1
and Q3. c, The circuit to implement the half-plaquette operations encodes the bit (ZZ) and phase (XX) parities of the twocode qubits’ Bell state |ψ〉 onto the respective syndrome qubits, Q2 (green) and Q4 (yellow). Arbitrary errors ε are intentionallyintroduced on the code qubit Q1 and detected from the correlated measurement of the syndrome qubits. Q2 (Q4) is initializedto |0〉 (|+〉 = (|0〉+ |1〉)/2). A Hadamard operation, H, is applied to Q4 before measurement.
each qubit are amplified by distinct Josephson parametricamplifiers (JPAs) giving high single-shot readout fidel-ity22,23. We implement two-qubit echo cross-resonancegates24, ECR = ZX90 − XI, which are primitives forconstructing controlled-NOT (CNOT) operations. Giventhe latticed structure of our device, we implement fourdifferent such gates, ECRij between qubits Qi (control)and Qj (target), with ij ∈ {12, 23, 34, 41}. In this ar-rangement, we use Q1 and Q3 (Fig. 1b, purple) as codequbits, Q2 as the Z−syndrome qubit (Fig. 1b, green)and Q4 as the X−syndrome qubit (Fig. 1b, yellow). AllECR gates are benchmarked24 with fidelities between0.93-0.97 (for further details see Methods). Single-qubitgates are benchmarked to fidelities above 0.998 with lessthan 0.001 reduction in fidelity due to crosstalk, as veri-fied via simultaneous randomized benchmarking25. Thefour independent single-shot readouts yield assignmentfidelities (see Methods) all above 0.94. These and otherrelevant system experimental parameters including qubitfrequencies, anharmonicities, energy relaxation and co-herence times are further discussed in the Methods.
To demonstrate the SC sub-lattice stabilizer measure-ment protocol (Fig. 1c), we first prepare the two codequbits in codeword state |ψ〉, which is a maximally en-tangled Bell state. Subsequently, the ZZ stabilizer isencoded onto the Z−syndrome qubit Q2, which is ini-tialized in the ground state |0〉. The XX stabilizer isencoded onto the X−syndrome qubit Q4, which is initial-
ized in the |+〉 = (|0〉+ |1〉)/√
2 state. Since we performmeasurements of the syndrome qubits in the Z measure-ment basis, Q4 also undergoes a Hadamard transforma-tion H right before measurement. The complete circuitas shown in Fig. 1c will detect an arbitrary single-qubiterror ε to the code qubits via the projective measure-ments of the syndrome qubits. We choose to apply theerror on Q1, but there is no loss of generality if appliedon Q3 instead. Each of the four possible outcomes of thesyndrome qubit measurements projects the code qubitsonto one of the four maximally entangled Bell states. Ifno error is present in the sequence, the syndrome qubitsare both found to be in their ground state after the meas-urement, and the prepared codeword state of the codequbits is preserved.
In our experiment, since the two code qubits (Q1 andQ3) are non-nearest neighbours in the lattice, the prepar-ation of the codeword state is performed via two-qubitinteractions with a shared neighbouring qubit, Q2. Thegate sequence for this state preparation can be compiledtogether with portions of the ZZ stabilizer encoding.The resulting complete gate decomposition of the circuitfrom Fig. 1c in terms of our available single and two-qubit ECR gates is described in detail in the Methodsand shown in Extended Data Fig. 3.
To implement arbitrary errors to the entangled codequbit state, we apply single-qubit rotations to Q1 of theform ε = Uθ, where U defines the rotation axis and θ
Only protects the state 1√2(|00〉+ |11〉) = 1√
2(|−−〉+ |++〉).
Offline correction (no feedback).
A.D. Corcoles et al., Nature Comm., 6, 6979, 2015.
TU Delft: phase-flip error correction
5 µm
Ancilla
Qubit 1
Qubit 2
Qubit 3
Optical
measurement
b c
d
c
CorrectDetect Detect
|±XiL
|0ia
0 10 20 30Time (ms)
0.5
0.75
1
Aver
age
stat
e fid
elity
to |±
Xi L
Best qubitLogical qubitError-corrected logical qubitNo feedback
X
Z
Z
Z
Time
Dataqubits:13CnuclearspinsAncillaqubit:NVcenterelectronspin
Only protects phase-flip errors: in these qubits phase-flip errors aredominant.
J. Cramer et al., ArXiv: 1508.01388v1.
Major obstacles to scaling
Fault-tolerant QEC: up to a hundred physical qubits per logicalqubit.
Each qubit benefiting from state of art properties: lifetime,coupling strength, tunability.
Avoid correlated errors: no undesired couplings.
Short time-scales for real-time feedback: feedback delays.
1 Classical and quantum noise, decoherence
2 Basics of quantum error correction
3 Recent experimental development: major obstacles
4 Some new directions
Directions followed by QUANTIC and collaborators
Hardware-efficient QEC: encode, protect and manipulatequantum information on a quantum harmonic oscillator whichbenefits from an infinite dimensional Hilbert space instead of amulti-qubit register.
Controllability: coupling the quantum harmonic oscillator to anancilla qubit provides controllability over its infinite dimensionalHilbert space.
Autonomous correction: replace real-time feedback by analogfeedback circuits ensuring the stabilization of a desired manifoldof quantum states by engineered dissipation.
Hardware-efficient QEC
Encode information on a Schrödinger cat state of storage cavity1.
Measure continuously photon-number parity as error syndrom2.
Perform error correction with real-time feedback and using the ancillaqubit3.
Storage
Readout
1 Z. Leghtas et al., Phys. Rev. A, 87: 042315, 2013.B. Vlastakis et al., Science 342: 607-610, 2013.
2 Z. Leghtas et al., Phys. Rev. Lett., 111, 120501, 2013.L. Sun et al., Nature 511, 444-448, 2014.
3 N. Ofek, A. Petrenko et al., In preparation.
Autonomous QEC
Engineer the dissipation of a quantum system by engineering itsinteraction with a reservoir to:
Stabilize an entangled state1.
Stabilize a manifold of quantum states2.
Perform autonomous QEC3.
System S Reservoir R
Engineered
interaction
dissipation
țHint
HR
Drives
HS
LETTERdoi:10.1038/nature12802
Autonomously stabilized entanglement between twosuperconducting quantum bitsS. Shankar1, M. Hatridge1, Z. Leghtas1, K. M. Sliwa1, A. Narla1, U. Vool1, S. M. Girvin1, L. Frunzio1, M. Mirrahimi1,2 & M. H. Devoret1
Quantum error correction codes are designed to protect an arbitrarystate of a multi-qubit register from decoherence-induced errors1, buttheir implementation is an outstanding challenge in the develop-ment of large-scale quantum computers. The first step is to stabilizea non-equilibrium state of a simple quantum system, such as a quan-tum bit (qubit) or a cavity mode, in the presence of decoherence. Thishas recently been accomplished using measurement-based feedbackschemes2–5. The next step is to prepare and stabilize a state of a com-posite system6–8. Here we demonstrate the stabilization of an entangledBell state of a quantum register of two superconducting qubits foran arbitrary time. Our result is achieved using an autonomous feed-back scheme that combines continuous drives along with a speci-fically engineered coupling between the two-qubit register and adissipative reservoir. Similar autonomous feedback techniques havebeen used for qubit reset9, single-qubit state stabilization10, and thecreation11 and stabilization6 of states of multipartite quantum systems.Unlike conventional, measurement-based schemes, the autonomousapproach uses engineered dissipation to counteract decoherence12–15,obviating the need for a complicated external feedback loop to cor-rect errors. Instead, the feedback loop is built into the Hamiltoniansuch that the steady state of the system in the presence of drives anddissipation is a Bell state, an essential building block for quantuminformation processing. Such autonomous schemes, which are broadlyapplicable to a variety of physical systems, as demonstrated by theaccompanying paper on trapped ion qubits16, will be an essentialtool for the implementation of quantum error correction.
Here we implement a proposal17, tailored to the circuit quantumelectrodynamics (cQED) architecture18, for stabilizing entanglementbetween two superconducting transmon qubits19. Each transmon con-sists of a Josephson junction capacitively shunted to form an anhar-monic oscillator whose lowest two levels are used as the qubit. Thequbits are dispersively coupled to an open cavity that acts as the dissi-pative reservoir. The cavity in our implementation is furthermore engi-neered to decay preferentially into a 50-V transmission line that wemonitor on demand. We show, using two-qubit quantum state tomo-graphy and high-fidelity single-shot readout, that the steady state of thesystem reaches the target Bell state with a fidelity of 67%, which is wellabove the 50% threshold that signifies entanglement. The fidelity canbe further improved by monitoring the cavity output and performingconditional tomography when the output indicates that the two qubitsare in the Bell state17. We implemented this protocol by means of post-selection and demonstrated that the fidelity increased to ,77%.
Our cQED set-up (Fig. 1a) consists of two individually addressablequbits, Alice and Bob, coupled dispersively to a three-dimensional (3D)rectangular copper cavity. The set-up is described by the dispersiveHamiltonian20
HB ~v0
Aa{azv0Bb{bzvgg
c c{c{12
aA(a{a)2
{12
aB(b{b)2{xAa{ac{c{xBb{bc{cð1Þ
Here a, b and c are respectively the annihilation operators of the Aliceand Bob qubits and the cavity mode; a{, b{ and c{ are the correspond-ing creation operators; v0
A and v0B are the Alice and Bob qubit angular
frequencies when there are no photons in the cavity; vggc is the cavity
frequency when both qubits are in the ground state (see Methods forexperiment parameters); aA and aB are respectively the Alice and Bob
1Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA. 2INRIA Paris-Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France.
JPC
AliceAlicea
Continuousdrives
Continuousdrives
BobBob
CavityCavity
OutputOutputInputInput
b
N
FBFA
Cavity spectrumCavity spectrum
ZB0
Z ZZ Z
FA
Qubit spectrumQubit spectrum
ZBnZA
0ZAn
c
... ...
|ee⟩
•••
FB
g ggg
eec
c cc
ee
|I+⟩
|gg⟩|0⟩ |1⟩ |n⟩
|ee⟩
|I+⟩|I–⟩
|gg⟩
|I–⟩
Figure 1 | Bell state stabilization set-up schematic and frequency landscapeof autonomous feedback loop. a, The qubits (Alice and Bob) are coupled tothe fundamental mode of a 3D cavity. Six continuous drives applied to thecavity input stabilize the Bell state | w2, 0æ. The cavity output jumps betweenlow and high amplitude depending on whether the qubits are in the desired Bellstate or not. The output is monitored by a quantum-limited amplifier (JPC).b, Spectra of the qubits and cavity coupled with nearly equal dispersive shifts(xA and xB); k is the cavity linewidth. Colours denote transitions that are drivento establish the autonomous feedback loop. c, Effective states of the systeminvolved in the feedback loop. Qubit states consist of the odd-parity states in theBell basis { | w2æ, | w1æ} and the even-parity computational states { | ggæ, | eeæ}.Cavity states, arrayed horizontally, are the photon-number basis kets | næ.Sinusoidal double lines represent the two cavity tones whose amplitudes createon average !n photons in the cavity when the qubits are in even-parity states.The cavity level populations are Poisson distributed with mean !n and we showonly | næ such that n<!n. Straight double lines represent four tones on qubittransitions. Collectively, the six tones and the cavity decay (decaying sinusoidallines) drive the system towards the ‘dark’ state, | w2, 0æ, which builds up asteady-state population.
0 0 M O N T H 2 0 1 3 | V O L 0 0 0 | N A T U R E | 1
Macmillan Publishers Limited. All rights reserved©2013
1 Z. Leghtas et al., Phys. Rev. A, 88: 023849, 2013.S. Shankar et al., Nature 504, 419-422, 2013.
2 M.M. et al., New J. of Physics, 16, 045014, 2014.Z. Leghtas et al., Science 347, 853-857, 2015.
3 J. Cohen and M.M., Phys. Rev. A, 90, 062344, 2014.