In these applications, quantum error correction is extremely important for protecting quantum states from decoherence. For example, measuring the state in (1.2) on the calculation basis will yield a 0, with probability |a|2, or a 1 with probability |b|2.
Introduction
Bit-flip code
The qubits are subject to bit rotation noise: that is, operators XII, IXI, IIX are the only possible errors. This procedure has two particularly attractive features: the error syndrome measurement does not distinguish between codewords, and the projective nature of the measurement discretizes all possible quantum errors into a finite set.
Bit-flip code in continuous time
The functions a(t)–d(t) express the probability that the system is left in a state that can be reached by zero, one, two, or three bits from the initial state, respectively. The overlap of this state with the initial state depends on the initial state, but is at least as large as when the initial state is |¯0i; that is, is at least as large as .
A true quantum error correcting code: the Shor code
It is possible to show [71] that this kind of indirect measurement can be described by Kraus operators [54] {Mr}, where r indexes the measurement result. In his construction of explanatory trees in [30], as we have seen, he considers only trees in which the number of cells in Shum (k) is almost the same as the span of the explanatory tree (k−1).
A quick look at the theory of quantum error correction
Stabilizer codes
The generators of such a group are a subgroup of this group such that each element of the stabilizer can be described as a product of the generators. Moreover, as we have seen, the stabilizer formalism gives a nice characterization of such code features as coded operators.
Topological quantum codes
Two-dimensional toric code
When a chain of faults (a series of links on the lattice or double lattice) occurs, the defects created arise at the endpoints, or boundary, of the chain on the lattice or double lattice. In that case, the entire operator acting on the system after the correction is simply part of the stabilizer and the coded state is not disturbed.
Local correction of toric codes
It can be analyzed by considering the dynamics of the measured system conditioned by the continuous measurement record; this process is referred to as unraveling. Had the result of the measurement not been known, the density matrix would have become the unconditional one.
Markovian feedback
Jump unravelings
Diffusive unravelings
Because of the white noise in the main stochastic equation (3.14), we call this diffusion untangling. In this case, the form of the master equation itself determines the large background jump rate, instead of the imposed classical field before detection.
State-estimate feedback
One-qubit picture
Before we show how the procedure for the bit-flip code works, we can get some intuition about how it works by considering an even simpler "code": the spin-up state (i.e., | 0i) of a single qubit. Note that if the Bloch vector lies exactly on the z-axis methZi<0, twisting it in one direction or another will move it to the spin-up state—the two directions are equivalent, and it is sufficient to turn one of them to choose arbitrarily.
Bit-flip code model
Decoherence (the γ term) shrinks the Bloch vector, measurement (the κ terms) lengthens the Bloch vector and moves it closer to the z axis, and correction (the λ term) rotates the Bloch vector in the y–z plane. Wherever the Bloch vector is in their zplane, the feedback forces it back to the spin-up state, which is the code space of this system.
Feedback for a general code
The first measure we used is the codeword fidelity Fcw(t) = tr(ρc(0)ρc(t)), the overlap of the condition with the target codeword. 4.6, using our protocol between discrete quantum error correction intervals of time 0.2/γ improves the reliability of the encoded data.
Relaxing assumptions
Bandwidth-limited control
This quantity corresponds to the fidelity of a state given continuous error correction up to γt=0.2, at which point discrete error correction is performed. The curves describing Fcw(t) for both sgn and tanh feedback are qualitatively similar: they both improve and exceed the fidelity of a single unprotected qubit, and ultimately exceed the fidelity that can be achieved by a round of discrete quantum error correction which is applied at time t.
Imperfect detection
The value figure here is the codeword fidelityFcw(t) = tr (ρc(0)ρc(t)), the overlap of the state with the target codeword. The analytical curves shown are as follows: the dotted line is the reliability of one decohering qubit, the dotted line is the reliability of three decohering qubits, and the dotted line is the reliability of a coded qubit after one round of discrete error correction at time t.
Conclusion
Example: A one-qubit toy model
In the frequency domain, this acts as a low-pass filter, and the output of this operation is thus a smoothed version of the high-frequency measurement current. We will use the signature of this smoothed measurement signal to infer the state of the system and thus condition the feedback.
Example: Bit flip correction
Again, we will assume that the errors on different qubits are independent and occur at the same error rate, and also that the measurement strength is the same for both stabilizer generators. As in the toy model, we need to smooth the measurement currents to obtain reliable error information.
Simulation results
The toy model
Here, |ψ(0)i is the initial state of the system, which is assumed to be |0i unless otherwise specified. The transitions of the expectation value of Z to −1 are due to errors, and the transitions back to +1 are due to feedback correction.
Three qubit code simulation
Both numerical results and heuristic analytical calculations show that the magnitude of the gauge strength should be roughly of the same order of magnitude as the Hamiltonian feedback strength for optimal correction. The third line in the graphs of Figure 5.2 shows the average reliability that can be achieved by discrete quantum error correction (using the same three qubit code) when the time between detection correction operations is t.
Inefficient measurement
The fact that the two curves largely intersect indicates that if the time between applications of discrete error correction is sufficiently large, a continuous protocol will maintain reliability better than a correspondingly discrete scheme. The thin solid curve is the reliability that can be achieved by discrete quantum error correction when the duration between applications is t,F3d(t).
Solid-state quantum computing with RF-SET readout
The probability that this happens is the measurement efficiency η(tM), which depends on the measurement time. A fault-tolerant procedure is to first measure all O(s4) qubits in a block of code.
Discussion and conclusion
Example: Spontaneous-emission correction
Two-qubit code: Jump unraveling
We consider a model in which the only decoherence process is due to spontaneous emission from statistically independent reservoirs. Following the presentation in Chapter 3, they are the jump operators for the spontaneous emission of the jth qubit. where 4κj is the decay rate for this qubit.
Two-qubit code: Diffusive unraveling
Such a code is optimal in the sense that it uses the smallest possible number of qubits required to perform the task of correcting a spontaneous emission error, since we know that the information stored in one uncoded qubit is spontaneous emission is destroyed.
Generalizations to n qubits
Since only one physical qubit is used in addition to the number of logical qubits, this is obviously again an optimal code. These schemes with a driving Hamiltonian do not have the admittedly desirable property of the given codes, in that if there is a time delay between the occurrence of the error and the application of the correction, the effective zero-emission evolution does not lead to additional errors. .
One-qubit general measurement operators
General unraveling
Note that we can now easily understand the error of the then hopping process of the spontaneous emission qubit considered in Sec. The generalization from spontaneous emission to the general jump operator cj for their case is straightforward: the code is the same as in the one-stabilizer protocol above, with a single stabilizer equal to (6.23).
Diffusive unraveling
We note that the work in this section can be easily modified to generalize the results of [43]. Therefore, if we are given the gauge operator e−iφc=χ+A+iB, we must choose a code with a stabilizer such that the condition (6.29) applies; then it is possible to find a reaction and a driving Hamiltonian such that the total evolution conserves the code space.
Diffusion as the limit of jumps
We saw in (6.33) and (6.34) that the feedback Hamiltonian needed to undo the effect of the jump operator enc+γ was fair. For simplicity, let's assume that the error rates on both qubits are the same (κ): furthermore, imagine that our value for κ is slightly off by a fraction ², so instead of correctly applying a driving Hamiltonian proportional to κ , apply a Hamiltonian proportional to κ(1 +²).
Universal quantum gates
Since the first term is proportional to the identity and the second term acts as zero on the code space, the effective evolution is given exclusively by Henc as long as the state remains in the code space under this evolution. Again, since D(1−S) annihilates the codespace, the effective evolution is given exclusively by Henc as long as the state remains in the codespace under this evolution.
Multiple channels
When there are two channels, a Bloch sphere analysis shows that it is possible to find a single S such that {S, D(j),α} = 0. Since it is possible to find a unit rotation which (1) take. toσZ as well as D(1),1 andD(1),2 to linear combinations ofσX andσY, this operator must anticommute with D(1),1 andD(1),2.
Looking at a spin 1 system
In the protocol given here we further assume that we know which error has occurred because measuring the error tells us which error has occurred. Unfortunately, this is a bit too simple to be true, as J+ operators do not fall into the category of operators that we are allowed to perform.
Conclusion
This means that at least 3L/4 of the surface must be inverted (since the minimum droplet surfaces can cover L/4 of the surface). With a success probability exponentially close to one, all errors in classical measured bits are removed.
The probability of failure for an encoded gate
An overview of Gacs’ proof of the fault-tolerance of Toom’s rule
The size of the cleanup circle dominated by the required measurements is O(s5) = O(log5L). The post-processing of the measured syndrome is performed level by level in the associated code.
Suppression of error droplets
The quantum problem: two-dimensional error droplets in four dimen-
As a result, cells in Noise in the obedient explanation share at least one edge with at least one cell in Noise in the disobedient explanation. It could be possible that the disobedient Explanation scores fewer points in Noise than the obedient one.
Quantum blowup without fast classical computation
Fault-tolerant measurement
It will contain several d/2-dimensional plaques of area 2d/2, both in the interior and on the surface of the coarse-grained cube. To do that, we first apply further fanouts to the measurement results, expanding the iteration code from lengths to lengths4(in depthO(logs4) =O(logs) and size O(s4)), and then we apply the conditional operation transversely. (in constant depth and sizeO(s4)), where each gate acts on a qubit of the 4-torus block depending on the value of the corresponding bit in the classical repetition code block.
Fault-tolerant fanout
Now we can see the purpose of the decoding step: for the protocol Ch, after the first step we actually have a 2h×2h number of bits. A corrupt block is one that has errors in at least half of the cells of the block.
Software preparation
If η is chosen to be greater than −logt, and if ²d is small enough, this expression will converge to no greater than 3²d, allowing us to use a fault-tolerant Von Neumann circuit. Note that if the repertoire of our noisy computer does not include all the gates used in the fault-tolerant simulation, a "Solovay-Kitaev" slowdown will appear in the construction of the necessary gates from the available gates.
Quantum blowup when classical computation is fast
Status of Pippenger’s conjecture
Efficient readout with a radio-frequency single-electron transistor in the presence of charge noise. Phys. Quantum Computing in the Presence of Spontaneous Emission Using a Combined Strategy of Dynamic Decoupling and Quantum Error Correction.
Stabilizer generators operators for the two-dimensional toric code
Homologically trivial and nontrivial cycles
Correction of an error chain
One-qubit feedback scheme
Feedback protocol with optimized feedback
Feedback protocol with non-optimized feedback
Feedback protocol behavior over a range of measurement and feedback strengths 41
Behavior of continuous plus discrete error correction
Feedback protocol with bandwidth-limited control
Behavior of feedback protocol for imperfect detection efficiency
Sample trajectory of one-qubit “code”
Behavior of filtered error correction protocol
Behavior of filter protocol for imperfect efficiency
Modified error correction protocol
A possible error path
A sample investigation graph with two histories
Illustrating the size
A spanned History with sub-Histories drawn
An Explanation Tree
Another error path
Another error path
