Hardware-efficient autonomous quantum error correction

0.77 MB·

by  Zaki Leghtas

Hardware-efficient autonomous quantum error correction
Zaki Leghtas,1 Gerhard Kirchmair,2 Brian Vlastakis,2 Robert
Schoelkopf,2 Michel Devoret,2 and Mazyar Mirrahimi1,2
1INRIA Paris-Rocquencourt, Domaine de Voluceau,
B.P. 105, 78153 Le Chesnay Cedex, France
2Department of Physics and Applied Physics,
Yale University, New Haven, Connecticut 06520, USA
(Dated: July 4, 2012)
Weproposeanewmethodtoautonomouslycorrectforerrorsofalogicalqubitinducedby
energyrelaxation. Thisschemeencodesthelogicalqubitasamulti-componentsuperposition
2
1 of coherent states in a harmonic oscillator, more specifically a cavity mode. The sequences
0
2 of encoding, decoding and correction operations employ the non-linearity provided by a

ul single physical qubit coupled to the cavity. We layout in detail how to implement these
J
operations in a practical system. This proposal directly addresses the task of building a
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hardware-efficient and technically realizable quantum memory.

]
h
p
-
I. INTRODUCTION
t
n
a
u
Long lived coherence is a prerequisite for quantum computation. The last two decades have
q
[
seen impressive improvements in the coherence times of qubits and cavities. The results of hard-

1
v ware improvement have been so substantial that the quality threshold needed for quantum error
9
7 correction (QEC) [1, 2] to be effective is within reach [3, 4]. Since the birth of QEC, many possible
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0 implementations have been proposed. We classify these QEC schemes as being either measurement
.
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based or autonomous. Measurement based QEC (MBQEC) consists of periodically measuring error
0
2
syndromes and feeding back appropriate correction pulses conditioned by the measurement results
1
:
v [5]. InautonomousQEC(AQEC),however, noclassicalinformationneedstobeextracted. Instead,
i
X
it is sufficient to transfer the random errors to an ancillary quantum system which is then reset
r
a
to remove the entropy [6, 7]. Furthermore, continuous time implementations can be used for both
MBQEC [8] and AQEC [9].
All these methods encode the single logical qubit to be protected in a register of several physical
qubits. While at least five physical qubits are needed to correct for single phase and bit flip errors
of the register [10], it has been shown that when the decoherence is due to a dominant quantum
noise process such as amplitude damping, less resources might be needed. Indeed, [11] propose a
2
four-qubit code correcting for single amplitude damping errors.
In this paper, we propose a QEC scheme which replaces the register of qubits by a single
high-Q cavity mode, coupled to a single physical qubit. The vastness of the Hilbert space of the
harmonic oscillator, combined with the control provided by operations described in [12], allows this
replacement. In our scheme, the logical qubit is encoded in a multi-component superposition of
coherent states in the cavity mode, the coupled qubit bringing the non-linearity necessary for the
manipulation of coherent states. This simple cavity-qubit system is the standard building block
of both circuit and cavity quantum electrodynamics (QED) experiments [13]. Here, we show that
this minimal hardware, together with an additional low-Q cavity mode used for qubit readout or
qubit reset, is sufficient to correct for the dominant source of errors, namely photon damping in the
high-Q cavity. Moreover, the number of independent quantum noise channels corrupting the logical
information, does not increase with the number of encoded qubits, which represents an additional
advantage of our protocol over a multi-qubit register. A cavity mode is thus a powerful hardware
for protecting quantum information [14–16].
Previous proposals [15, 16] have suggested that states with a given photon number parity of a
cavity mode (parity states) can be used to encode quantum information. There, stabilization of a
parity manifold is obtained by a quantum non-demolition (QND) parity measurement and photon
injectionwhenneeded. Suchaschemereplacesthedecoherenceduetophotondampingbyaslower,
unusual dephasing due to a drift in the parity manifold.
Here, wegoastepforwardtowardsareadilyrealizablequantummemory, byproposingascheme
for efficiently encoding a logical qubit in a particular cavity state that is fully protected against
single quantum jumps due to photon damping. Depending on the experimental constraints, two
approaches are possible. The first approach, a MBQEC scheme, keeps track of the quantum jumps
by stroboscopic QND measurements of the photon number parity and corrects for the decay by an
appropriate decoding and encoding operation sequence. The second approach, an AQEC scheme,
transfers the entropy of the cavity state to an ancilla qubit and then removes this entropy by
reseting the qubit state. All the encoding, decoding and correction operations of both approaches
can be performed using tools that have been introduced in our recent paper [12].
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II. CAVITY LOGICAL 1 AND LOGICAL 0, AND MBQEC
An arbitrary qubit state c |g(cid:105)+c |e(cid:105) (we denote |g(cid:105) and |e(cid:105) the ground and excited state) is
g e
mapped into a multi-component coherent state (cid:12)(cid:12)(cid:12)ψα(0)(cid:69) = cg|Cα+(cid:105)+ce(cid:12)(cid:12)Ci+α(cid:11), where
(cid:12)(cid:12)C±(cid:11) = N(|α(cid:105)±|−α(cid:105)) , (cid:12)(cid:12)C±(cid:11) = N(|iα(cid:105)±|−iα(cid:105)) .
α iα
√
N (≈ 1/ 2) is a normalizing factor and |α(cid:105) denotes a coherent state of complex amplitude α,
(cid:12) (cid:69)
chosen such that |α(cid:105),|−α(cid:105),|iα(cid:105),|−iα(cid:105) are quasi-orthogonal. Together with (cid:12)ψ(0) , we introduce
(cid:12) α
(cid:12)(cid:12)(cid:12)ψα(1)(cid:69) = cg(cid:12)(cid:12)Cα−(cid:11)+ice(cid:12)(cid:12)Ci−α(cid:11)
(cid:12)(cid:12)(cid:12)ψα(2)(cid:69) = cg(cid:12)(cid:12)Cα+(cid:11)−ce(cid:12)(cid:12)Ci+α(cid:11)
(cid:12)(cid:12)(cid:12)ψα(3)(cid:69) = cg(cid:12)(cid:12)Cα−(cid:11)−ice(cid:12)(cid:12)Ci−α(cid:11) .
(cid:12) (cid:69)
We have recently proposed a toolbox of operations which efficiently prepare the states (cid:12)ψ(n) [12].
(cid:12) α
The logical 0, |C+(cid:105), and the logical 1, (cid:12)(cid:12)C+(cid:11), have the three following remarkable properties: first,
α iα
(cid:12) (cid:69)
the states (cid:12)ψ(n) evolve after a quantum jump due to a photon loss, to
(cid:12) α
(cid:12) (cid:69) (cid:13) (cid:12) (cid:69)(cid:13) (cid:12) (cid:69)
a(cid:12)ψ(n) /(cid:13)a(cid:12)ψ(n) (cid:13) = (cid:12)ψ((n+1)mod4) ,
(cid:12) α (cid:13) (cid:12) α (cid:13) (cid:12) α
(cid:12) (cid:69)
where a is the annihilation operator. Therefore the set {(cid:12)ψ(n) } is closed under the action of a.
(cid:12) α
(cid:12) (cid:69) (cid:12) (cid:69)
Second, in absence of jumps during a time interval t, (cid:12)ψ(n) deterministically evolves to (cid:12)ψ(n) ,
(cid:12) α (cid:12) αe−κt/2
where κ is the cavity decay rate. Third, defining the parity operator Π = exp(iπa†a), we have
(cid:68) (cid:12) (cid:12) (cid:69)
ψ(n)(cid:12)Π(cid:12)ψ(n) = (−1)n .
α (cid:12) (cid:12) α
The parity operator acts therefore as a quantum jump indicator. Now, suppose we have a QND
parity measurement, and that we have counted c jumps during a time t, the initial state has
(cid:12) (cid:69)
evolved to (cid:12)ψ(cmod4) . Using operations introduced in [12], we can find a unitary transformation,
(cid:12) αe−κt/2
(cid:12) (cid:69) (cid:12) (cid:69)
independent of c and c , which maps (cid:12)ψ(cmod4) back to (cid:12)ψ(0) , therefore undoing the effect of
g e (cid:12) αe−κt/2 (cid:12) α
decoherence. How to find this correcting transformation will be discussed in section IV.
This MBQEC we have outlined so far requires a high throughput QND parity measurement and
a low latency feedback loop. Let us note that QND parity measurements have been previously
performed through Ramsey-type experiments within the context of cavity QED with Rydberg
atoms [17]. Such a measurement scheme can be also adapted to circuit QED experiments but
necessitates a flux bias line which would allow one to alternately tune the qubit near or far from
resonancewiththecavitymode. Also, fastandreliablemeasurementswouldnecessitateapplication
4
of quantum limited amplifiers. In the next section, we introduce an AQEC scheme which does
not necessitate such resources. Instead, it requires the availability of a rapid, high fidelity qubit
reset [18].
III. AUTONOMOUS QEC
AQECisrealizedbyusinganauxiliaryquantumsystemthatwetakeheretobethesamecoupled
physical qubit, which is used to manipulate the cavity state. The idea consists in finding a unitary
operation U such that
correct
Ucorrect :|g(cid:105)⊗(cid:12)(cid:12)(cid:12)Cα±e−κt/2(cid:69) → √12(|g(cid:105)±|e(cid:105))⊗(cid:12)(cid:12)Cα+(cid:11), (1)
|g(cid:105)⊗(cid:12)(cid:12)C± (cid:69) → √1 (|g(cid:105)±|e(cid:105))⊗(cid:12)(cid:12)C+(cid:11).
(cid:12) iαe−κt/2 2 iα
Thisunitaryoperationtransferstheentropyofthequantumsystemtobeprotectedtotheauxiliary
one. Now, resetting the state of the auxiliary system, we can evacuate the entropy, restoring the
initial full state.
(cid:12) (cid:69)
Here, the AQEC scheme consists of encoding the qubit state c |g(cid:105)+c |e(cid:105) in the state (cid:12)ψ(0)
g e (cid:12) α
and, in a stroboscopic manner, performing the above unitary transformation followed by the qubit
reset. Assuming that at most one quantum jump can happen between two correction operations
(cid:12) (cid:69) (cid:12) (cid:69)
separated by time T , the state before the correction is given either by (cid:12)ψ(0) or (cid:12)ψ(1) .
w (cid:12) αe−κTw/2 (cid:12) αe−κTw/2
(cid:12) (cid:69)
After the correction operation, we have restored the initial state (cid:12)ψ(0) . This whole process may be
(cid:12) α
finished by a decoding step transferring the quantum information back onto the qubit (see Fig. 1).
(n)
We now quantify the performance of our AQEC scheme. Let ρ denote the projector onto the
α
(cid:12) (cid:69)
state(cid:12)ψ(n) . TheeffectofthewaitingtimeT betweentwocorrectionsmaybemodeledbyaKraus
(cid:12) α w
operator
K : ρ(0) → p ρ(0)+p ρ(1)+p ρ(2)+p ρ(3) ,
w α 0 α˜ 1 α˜ 2 α˜ 3 α˜
where α˜ = αe−κTw/2. For a Poisson process with a jump rate λjump, the probability of hav-
ing k jumps during a time interval T is given by exp(−λ T )λk Tk/k!. We denote p
w jump w jump w k
the probability of having k (mod 4) jumps during the waiting time T . In the limit where
w
(cid:15) ≡ λ T = κT n¯ (cid:28) 1, we have p ≈ 1 − (cid:15) + (cid:15)2 /2, p ≈ (cid:15) − (cid:15)2 ,
jump jump w w 0 jump jump 1 jump jump
p + p ≈ (cid:15)2 /2. The correction step consists of the joint unitary operation on the cavity-
2 3 jump
qubit system followed by the qubit reset. We model the effect of this operation by the Kraus
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1 2 3 4
1 error
or -
+
0 error +
e t e
d c d
o e o
c rr c
n o e
E C D
QEC cycle
FIG. 1: Our AQEC scheme is composed of three operations: encoding, correcting and decoding. The
joint cavity-qubit state is represented by a generalized Fresnel diagram. Fresnel plane positions carry
the description of the cavity mode, while colors carry the description of the qubit. Our protocol only
requires that we represent superpositions of coherent states entangled with the qubit degrees of free-
dom. A circle whose center is positioned at α in the diagram corresponds to a coherent state compo-
nent of amplitude α. For example, the diagram in frame 2 represents the state cg|Cα+(cid:105) + ce(cid:12)(cid:12)Ci+α(cid:11) =
√
N (c |g,α(cid:105)+c |g,−α(cid:105)+c |g,iα(cid:105)+c |g,−iα(cid:105)), where N ≈1/ 2 is a normalization factor. Each compo-
g g e e
nent of this state corresponds to a circle whose color refers to whether the qubit is in |g(cid:105) (blue) or |e(cid:105) (red).
The rim of each circle indicates whether the pre-factor is c (single line) or c (double line). Finally, the
g e
fraction of the colored disc represents the total weight |Nc |2 of each coherent component. Here, quarter
g,e
filled circles correspond to |Nc |2 = 1/4. Initially (frame 1), the qubit is in c |g(cid:105)+c |e(cid:105) and the cavity
g,e g e
is in vacuum. The plus [resp: minus] sign in the 2 and 3 diagrams indicates whether the logical qubit is
encoded in the pair (|C+(cid:105),(cid:12)(cid:12)C+(cid:11)) [resp: (|C−(cid:105),i(cid:12)(cid:12)C−(cid:11))]. A jump from a plus to a minus sign is induced by
α iα α iα
a photon loss error, which we aim to correct.
(cid:12) (cid:69) (cid:12) (cid:69) (cid:12) (cid:69)
operator K , mapping both (cid:12)ψ(0) and (cid:12)ψ(1) to (cid:12)ψ(0) . After N correction cycles and
c (cid:12) αe−κTw/2 (cid:12) αe−κTw/2 (cid:12) α
waiting times (each one taking a time T + T ), we obtain a fidelity at time t = N(T + T ):
c w N c w
(cid:12)(cid:68) (cid:12) (cid:12) (cid:69)(cid:12)2
F (t ) = (cid:12) ψ(0)(cid:12)(K K )N (cid:12)ψ(0) (cid:12) .
AQEC N (cid:12) α (cid:12) c w (cid:12) α (cid:12)
We denote (1 − (cid:15) ) the fidelity of the correction operation, taking into account various
correct
imperfections and particularly finite coherence times. Also, (cid:15) = (cid:15)2 /2 denotes the probabil-
wait jump
ity of having 2 or more jumps during the waiting time between two correction steps. We have
F (t ) ≈ ((1−(cid:15) )(1−(cid:15) ))N. Assuming T (cid:28) T , we obtain an effective decay rate
AQEC N correct wait c w
√
κ ≈ ((cid:15) +(κT n¯)2/2)/T . The latter is maximal for T = 2(cid:15) /κn¯, which would lead
eff correct w w w correct
to
√
κ = κn¯ 2(cid:15) . (2)
eff correct
√ (cid:12) (cid:69)
This is an improvement by a factor of 2(cid:15) with respect to the decay rate κn¯ of (cid:12)ψ(0) in
correct (cid:12) α
absence of correction.
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IV. ENCODING, DECODING AND CORRECTING OPERATIONS
(a) Encode
c |C+(cid:105)
|0(cid:105) Dα Π2 D-iα Dβ Π2 D-iβ Π2 D-β D-α +gce(cid:12)(cid:12)αCi+α(cid:11)
c |g(cid:105)
g • X0π • X0π • X0 X0 |g(cid:105)
- ,0 ,0 -π,2n¯ -π,2n¯
+c |e(cid:105) 2 2
e
(b) Decode
c |C+(cid:105)
+gce(cid:12)(cid:12)αCi+α(cid:11) Dα Diβ Π2 Dβ Π2 D-iβ D-iα Π2 D-α |0(cid:105)
c |g(cid:105)
|g(cid:105) X0 • X0 • X0π X0π • g
π,0 π,-2n¯ - ,2n¯ ,0
2 2 +c |e(cid:105)
e
FIG. 2: Sequence of operations which generate U (a) and U (b), mapping the qubit state to the
encode decode
cavity and back. D displaces the cavity state by an amplitude α regardless of the qubit state. Conditional
α
operation Π [resp: Π] is realized by simply waiting for time π/χ [resp: π/(2χ)]. This transforms states of
2
the form |e,α(cid:105) to |e,−α(cid:105) [resp |e,iα(cid:105)], and leaves |g,α(cid:105) unchanged. The conditional qubit rotation X0
θ,η
rotatesthequbitstatebyeθ2(eiη|e(cid:105)(cid:104)g|−e−iη|g(cid:105)(cid:104)e|) onlyifthecavityisinthevacuumstate|0(cid:105). Thisisachieved
by applying a long selective pulse exploiting the energy level dispersive shifts. We denote β = α(−1+i)
and n¯ =|α|2.
We define the unitary operations U and U such that for all c and c ,
encode decode g e
Uencode :(cg|g(cid:105)+ce|e(cid:105))⊗|0(cid:105) → |g(cid:105)⊗(cg(cid:12)(cid:12)Cα+(cid:11)+ce(cid:12)(cid:12)Ci+α(cid:11)) .
Udecode :|g(cid:105)⊗(cg(cid:12)(cid:12)Cα+(cid:11)+ce(cid:12)(cid:12)Ci+α(cid:11)) → (cg|g(cid:105)+ce|e(cid:105))⊗|0(cid:105).
In this section, we show how we could perform U , U and the correcting operation
encode decode
(1), in practice. We place ourselves in the strong dispersive regime, where both the qubit and
the resonator transition frequencies split into well-resolved spectral lines indexed by the number
of excitations in the qubit and the resonator [19]. The resonator frequency ω splits into two well
r
resolved lines ωg and ωe, corresponding to the cavity’s frequency when the qubit is in the ground
r r
(|g(cid:105)) or the excited (|e(cid:105)) state. Through the same mechanism, the qubit frequency ω splits into
q
{ωn} corresponding to the qubit frequency when the cavity is in the photon number state
q n=0,1,2,···
|n(cid:105). Recent experiments have shown dispersive shifts that are more than 3 orders of magnitude
larger than the qubit and cavity linewidths [3].
7
(a) Encode
Dα Π2 D-iα X-0π,0 Dβ
2
Π,X0 D X0 D
2 π,0 -β -π,2n¯ -α
2
D-iβ Π2,X-0π,2n¯
(b) Decode
Dα Xπ0,0 Diβ Π2,Xπ0,-2n¯ Dβ
Π2,X-0π,2n¯ D-iα Π2 D-α
2
D-iβ X0π,0
2
FIG. 3: Diagrammatic equivalent of Fig. 2. The frames are ordered from left to right and top to bottom.
The diagram notations are explained in Fig. 1. The symbol given in frame n corresponds to the operation
performed to go from frame n−1 to n. All the possible operations are described in the caption of Fig. 2.
The curved arrow corresponds to the rotation of the excited state component of the state.
1
The Hamiltonian of such a dispersively coupled qubit-cavity system is well approximated by
σ σ
H = ω z +ω a†a−χ za†a ,
0 q c
2 2
where ω and ω are respectively the qubit and cavity frequencies, χ is the dispersive coupling
q c
and σ = |e(cid:105)(cid:104)e| − |g(cid:105)(cid:104)g|. This Hamiltonian may be written in an appropriate rotating frame
z
as H = −χ|e(cid:105)(cid:104)e|a†a. This dispersive coupling is called strong when χ (cid:29) κ,1/T , where T =
2 2
(1/2T +1/T )−1 is the qubit decoherence time.
1 φ
As detailed in [12], the strong dispersive cavity-qubit coupling allows to efficiently perform
1
conditional cavity displacements and conditional qubit rotations. Long selective qubit pulses with
carrier frequency ω0 can rotate the qubit state conditioned on the cavity being in the vacuum
q
state. Similarly, selective cavity pulses with carrier frequency ωg [resp: ωe] can coherently displace
r r
the cavity state conditioned to the qubit being in the ground [resp: excited] state. Furthermore,
as explained in [12, 20], shorter operation times are obtained by replacing a conditional cavity
8
(a) Transferring entropy from the cavity to the qubit, and qubit reset
c |α(cid:48)(1-i)(cid:105)
g
π
+i1∓21ccge(cid:12)(cid:12)C(cid:12)(cid:12)Cα±i±(cid:48)α(cid:11)(cid:48)(cid:11) Diα(cid:48) D-β(cid:48) Π2 Diβ(cid:48) Π2 +ceei4ein¯(cid:48)|0(cid:105)
|g(cid:105) X0 • X0 • X0π π Reset
π,-2n¯(cid:48) π,-2n¯(cid:48) 2,2 |g(cid:105)±|e(cid:105) |g(cid:105)
√
2
(b) Re-pumping energy into the cavity
c |α(cid:48)(1-i)(cid:105) c |α(1-i)(cid:105)
+gceeiπ4ein¯(cid:48)|0(cid:105) D-βd(cid:48) Π D-βd(cid:48) g +ceein¯|0(cid:105)
|g(cid:105) X0 π • X0 |g(cid:105)
π,n¯-n¯(cid:48)-4 -π,0
(c) Re-encoding the logical qubit into the cavity logical 0 and 1 states
c |α(1-i)(cid:105) c |C+(cid:105)
+gceein¯|0(cid:105) Dβ Π2 D-iβ Π2 D-β D-α +gce(cid:12)(cid:12)αCi+α(cid:11)
|g(cid:105) X0π • X0π • X0 X0 |g(cid:105)
,0 ,0 -π,2n¯ -π,2n¯
2 2
FIG. 4: Full correcting sequence obtained by concatenating the three sequences of pulses (a-c). (a) First,
the entropy is transferred from the cavity to the qubit, and then, the qubit is reset to its ground state. (b)
Energy is re-pumped into the coherent component to compensate the deterministic decay due to damping
during the waiting time T between two correction sequences. (c) The cavity state is mapped back onto
w
the initial cavity logical 0 and logical 1. See the caption of Fig. 2 for a description of operations D ,Π
α
and Xθ0,η. Here, we denote α(cid:48) = e−κTw/2α the damped amplitude after the waiting time Tw, n¯(cid:48) = |α(cid:48)|2
and β(cid:48) = α(cid:48)(i−1). In order to compensate the damping during T , during the re-pumping step, we take
w
β(cid:48) =(β(cid:48)−β)/2.
d
displacement by two unconditional ones separated by a waiting time.
The operations involved in our QEC scheme rely on a qubit reset and three unitary transfor-
mations. The first one, D , displaces the cavity state by a complex amplitude α regardless of the
α
qubit state. Second, the conditional operation Π [resp: Π] transforms states of the form |e,α(cid:105) to
2
|e,−α(cid:105)[resp|e,iα(cid:105)],andleaves|g,α(cid:105)unchanged. Itisrealizedbysimplywaitingfortimeπ/χ[resp:
θ
π/(2χ)]. Third, a conditional qubit rotation X0 rotates the qubit state by e2(eiη|e(cid:105)(cid:104)g|−e−iη|g(cid:105)(cid:104)e|)
θ,η
only if the cavity is in the vacuum state |0(cid:105). This is achieved by applying a long selective pulse
exploiting the energy level dispersive shifts.
9
(a) Transferring entropy from the cavity to the qubit, and qubit reset
D Π,X0 D
-β′ 2 π,-2n¯′ iβ′
D X0
iα′ π,-2n¯′
Π X0 qubit reset
2 π,π
2 2
(b) Re-pumping energy into the cavity
Xπ0,n¯-n¯′-π4 D-βd′ Π D-βd′ X-0π,0
(c) Re-encoding the logical qubit into the cavity logical 0 and 1 states
X0 D Π,X0
π,0 β 2 π,0
2 2
D Π,X0
-iβ 2 -π,2n¯
D X0 D
-β -π,2n¯ -α
1
FIG. 5: Diagrammatic equivalent of Fig. 4. See Fig. 1 for an explanation of the diagram notation. In the
frame before last of (a), the error is encoded in the phase of the qubit superposition, and is not represented
in this diagram. After qubit reset (last of frame of (a)), this phase information is erased.
Theresetoperationforcesthequbitstateto|g(cid:105)independentlyofthecavitystate. Thisoperation
needstobefastcomparedtoχtoavoidre-entanglementofthequbittothecavitymode. Apossible
1
schemetoperformsuchafastresetistorapidlytune(e.g. withafluxbiasline)thequbitfrequency
tobringitintoresonancewithalow-Qcavitymode[18]. Anotherpossibility,avoidingfastfrequency
tuning, is to perform a dynamical cooling cycle as proposed in [21]. See Fig. 2 and Fig. 4 for a
detailed illustration of how combining all these operations leads to the encoding, decoding and
correcting gates. A graphical representation of the sequence of operations is given in Fig. 3 and
1
10
Fig. 5. In the correction sequence (see Figs. 4, 5), we have introduced the qubit reset in the middle,
in contrast with what Eq. (1) suggests. We find the resulting sequence to be more efficient.
Simply adapting the sequence of Fig. 4, without the qubit reset, we would obtain the correcting
sequence suggested in the MBQEC scheme of section II.
V. SIMULATIONS
100
80
) AQEC
%
( 60 no AQEC

y
Bare qubit in |e>
t
i
l
e 1 photon in bare cavity
d
i 40
F
20
0
0 200 400 600 800
Time (µ s)
(cid:12) (cid:69)
FIG. 6: Fidelity of our AQEC scheme. The cavity state is initialized in state (cid:12)ψ(0) . We represent the
(cid:12) α
(cid:12) (cid:69)
fidelity of the cavity state w.r.t (cid:12)ψ(0) in absence of the error correction sequence of Fig. 4 (green solid line)
(cid:12) α
and after each correction sequence (blue open dots). These simulations take into account dominant sources
of decoherence: cavity decay T = 2 ms and qubit T = T = 100 µs. The decay rate of the bare qubit
cav 1 2
and cavity are plotted for comparison (red and magenta dashed lines).
To numerically compute the fidelity of each operation, we simulate the corresponding sequence
of pulses on a master equation taking into account the qubit and cavity decoherence. We take
a qubit decay time T and pure dephasing time T of T = T = 100 µs and a cavity lifetime
1 2 1 2
T = 1/κ = 2 ms. The dispersive coupling strength is χ/2π = 40 MHz and the mean photon
cav
number per coherent component is n¯ = |α|2 = 4. We get an encoding and decoding fidelity of
1 − (cid:15) = 1 − (cid:15) = 99.65% for an operation time of T = T = 231ns, and a correcting
encode decode e d
fidelity of 1−(cid:15) = 99.23% for an operation time of T = 519 ns. The optimal waiting time is
correct c
T = 65.6 µs. This leads to an effective lifetime for the corrected state of Teff = 4.1 ms, within 1%
w cav