In this section, we will first introduce the distance-2 surface code block that will be used to demonstrate the continuous feedback scheme. Then, we will briefly review key concepts of the SLH dynamical model before deriving the detail model of our surface code in the continuous measurement feedback scheme.
3.2.1 Distance-2 Surface Code
The surface code lattice that we will consider in the following sections comprises of four data qubits and five syndrome measurements in a 2×2 configuration as shown in Fig. 3.2. The code stabilizer generators areZ1Z2Z3Z4,X1X2,X1X3,X2X4, andX3X4.
1
2
3
4
Z
X
X
X
X
Fig. 3.2 Distance-2 surface code
This code lattice has a code distance of two which is insufficient to correct arbitrary error such as the general depolarizing error: ρ7→(1−p)ρ+3p(XρX+YρY+ZρZ).
Indeed,X errors cannot be localized by this lattice since all fourX errors have the same syndrome. However, single-qubitY andZ errors can be identified by combining the five syndrome measurements.
Unlike discrete QEC, the continuous syndrome measurement does not require a physical qubit but only a field interacting with the relevant data qubits with some specific coupling parameters. The output fields will carry both information about the syndrome,
3.2 Model 57
as well as additional noise. This information can be used to estimate the original syndrome. This input-output process as well as the filter and estimation will be described by the quantum stochastic differential equations.
3.2.2 Continuous QEC in the SLH Framework
Our continuous QEC feedback network will be described in the Heisenberg-picture using the Hudson-Parthasarathy quantum stochastic differential equation (QSDE) [79], which is also known as the input-output formalism [49] in the physics literature. The QEC network is encapsulated by the SLH parametrization [56], which comprises of scattering matrixS, coupling vectorL, and HamiltonianH. Unless otherwise stated, we assume there is no scattering between quantum fields, soS=I.
In the continuous QEC scheme, our surface code is coupled to two separate groups of channels. The first is a collection of error channels (LE) which depend on the chosen error model. For generic balanced depolarizing channels, the error coupling vector for a collection ofN qubits with per-qubit error rate ofγ is:
LE = r γ 3σ (j) i j∈[1,N] i=x,y,z .
This corresponds to a per time step error probability ofγdt for each qubit.
As noted before, we will use a limited error model for our distance-2 surface code that only includesY andZ errors but notX. There are therefore a total of eight error channels coupling to the surface code lattice in Fig. 3.2. Despite having separate channels for errors, we will assume that all of these are unobservable. Indeed, if we can access the error channels, there will be no need for syndromes as errors can be corrected directly.
In order to measure the error syndromes, we need to create five syndrome field channels with equal coupling strength, denoted byκ:
58 Chapter 3. Continuous Quantum Error Correction
Surface Code
Array Estimator Controller
dAS,LS
dAE,LE dAE,out
dAS,out Sˆ(t) F(t)
Fig. 3.3 Block diagram of surface code continuous error correction in the SLH framework: the surface code array is coupled to two groups of channels, namely error channels (dAE) and syndrome channels (dAS). The coupling strengths areLE andLS, respectively. The syndrome outputs are measured to estimate the syndrome conditional expectation values ( ˆS(t)). The Hamiltonian feedback (F(t)) is a function of the syndrome estimators.
The unitary evolution of the surface code network can be derived using the Hudson- Parthasarathy QSDE: dU(t) =ndA†E(t)LE−L†EdAE(t)−1 2L † ELEdt +dA†S(t)LS−L†SdAS(t)−1 2L † SLSdt o U(t), (3.1)
where we assume no internal dynamics (H=0). The QSDE for an arbitrary observable
X on the qubit network is given by:
d jt(X) = jt(LLE(X) +LLS(X))dt
+jt([L†E,X])dAE(t) + jt([X,LE])dA†E(t) +jt([L†S,X])dAS(t) + jt([X,LS])dA†S(t).
(3.2)
The measurement signals are taken to be of the forms of homodyme detection on the syndrome field channels, i.e.dY(t) =dAS(out)(t) +dA†S(out)(t).
Assuming that the input fields are in vacuum state, the five output equations can be explicitly derived for the surface code lattice in Fig. 3.2:
3.2 Model 59
wheredWi(t) =dASi(t) +dA†Si(t)is equivalent to a classical Wiener process2, that is a Gaussian distributed random variable with zero mean and variance ofdt.
By observing the output fielddYi(t), we can write down the optimal estimate of an system observable using quantum filtering techniques:
dπt(X) = πt LLE(X) +LLS(X)dt (3.4)
+
∑
i
{πt({LSi,X})−2πt(LSi)πt(X)}dWi(t),
where we have used the Hermitian property of syndrome operators (LSi=L†Si) to simplify the equation. The notationπt(X)stands for the conditional expectation of the observable X given all measurement records up to timet.
So far, we have not made any simplification by using assumptions about the error model or observable operators. Despite using a limited error model for our distance-2 lattice, we derive the filtering estimation equations for general depolarizing noise as this will definitely be used for larger lattice with full error correction capability.
Denote{gi}the set of syndrome generators, by using (3.4) we can get their filtering equations: dπt(gl) =−(4w)γ 3 πt(gl)dt +2√κ k
∑
i=1 {πt(glgi)−πt(gl)πt(gi)}dWi, (3.5)wherewdenotes the Pauli weight of the parity operatorgl.
The filtering equations for syndrome measurements are non-linear due to the appear- ance of high-order terms likeπt(gl)πt(gi). Also, they expand beyond the initial stabilizer
generator set because of operator product terms likeπt(glgi). However, they will form a finite set of equations which include all operators generated by the group of syndrome generators.
Having the estimation of the syndrome state, we can proceed to the next step to build an estimation-based feedback controller to correct the surface code lattice as diagrammatically shown in Fig. 3.3.
60 Chapter 3. Continuous Quantum Error Correction