4.3 Measurement-Based Quantum Computation
4.3.2 The Model-Conversion Lemma
At first, it seems that the analysis of noisy measurement-based quantum computation is complicated by the fact that the measurement bases within each gate simulation depend on the outcomes of previous measurements. For example, if one measurement outcome is erroneous, then this will lead to adapting incorrectly some future measurement bases; hence, one fault can cause a series of other
correlated faults. In order to obtain a proof of the quantum threshold theorem for measurement- based models, we will therefore need to show that such correlated faults are not an obstacle to executing fault-tolerant quantum computation.
Before proceeding further, it is helpful to develop a formalism of describing individual gate sim- ulations in measurement-based models and how simulations compose together. For every simulated gate, let us define alevel-0 simulator, or0-Sim, as the union of operations required for simulating this gate. For example, the 0-Sim corresponding to the Hadamard gate consists of the preparation of an ancillary qubit in the state|+i, acphasegate and a measurement along the eigenbasis ofX (Fig. 4.5)—note that the conditional gate X is not contained in the 0-Sim since Pauli corrections are never directly applied.
As discussed previously, every gate simulation in measurement-based models is achieved up to known Pauli corrections that are recorded in a classical memory and are used to adapt future measurement bases. To make the role of Pauli corrections more transparent, we will define anideal Pauli corrector, ori-corrector. An i-corrector is an ideal (i.e., faultless) device that conditioned on the classical record of the Pauli correction of a given qubit, applies this Pauli correction operator and discards the classical record. Like i-decoders and i-LRUs, i-correctors will be tools for our analysis and do not correspond to physical operations applied during a measurement-based computation. If we use a double line to denote theclassicalbits carrying the information about the Pauli correction of some qubit, then the i-corrector acting on this qubit is the operation
i-corrector = • P
,
whereP denotes the Pauli correction operator that the classical record indicates. We can now state what it means for a 0-Sim to correctly simulate some 0-Ga.
Definition 11 (Correctness for 0-Sims). A 0-Sim iscorrectif the 0-Sim followed by i-correctors is equivalent to i-correctors followed by the ideal 0-Ga that the 0-Sim simulates. For a single-qubit gate
0-Ga, correctness means schematically:
correct
0-Sim i-corrector = i-corrector ideal
0-Ga
.
We will also say that
Definition 12 (Goodness for 0-Sims). A 0-Sim isgoodif it containsnofaults, otherwise it is bad.
Since a good 0-Sim contains no faults, it operates ideally; hence, the following lemma holds.
Lemma 6(0-Sim-Correct). A good 0-Sim is correct.
It is sufficient to construct 0-Sims for every gate 0-Ga in a finite universal gate set; e.g., such constructions for the graph-state model were discussed in the previous section. Definitions 11 and 12 also hold for 0-Sims simulating qubit preparation or measurement 0-Gas. The correctness of good measurement 0-Sims allows us to replace them by i-correctors followed by the corresponding ideal measurement 0-Gas, thereby creating i-correctors. Similarly, the correctness of good preparation 0-Sims allows us to annihilate i-correctors together with preparation 0-Sims, thereby obtaining the ideal preparation 0-Gas alone. We can now state our basic lemma.
Lemma 7(Model Conversion). Consider the measurement-based simulation of operations in a finite quantum universal set, G, and let Csim be the maximum number of locations in any 0-Sim. Then, if the measurement-based simulation of a quantum circuit expressed in terms of operations in G is subject to local noise of strength ε, there is a quantum circuit with operations inG which produces the same output probability distribution and is subject to local noise with strength
ˆ
ε≤Csimε
(orpˆ≤Csimpif noise is stochastic).
The proof of this lemma is essentially the same as the proof of the leakage-reduction lemma and will not be discussed in detail. Similar to coherent i-LRUs and their inverses, we can consider the coherent version of an i-corrector which retains the classical record of Pauli corrections. We can insert pairs of coherent i-correctors and their inverses preceding every bad 0-Sim in order to convert them to the simulation of some faulty operations (which may be correlated since they depend on the classical record of the Pauli corrections). Overall, by creating i-correctors out of 0-Sims simulating measurements, moving them to the left of 0-Sims simulating quantum gates, and annihilating them inside 0-Sims simulating qubit preparations, we can convert any noisy measurement-based simulation of an ideal quantum circuit to a direct noisy implementation of the quantum circuit. Then, local
noise afflicting the measurement-based simulation will be equivalent to local noise acting on the equivalent quantum circuit, and the transformation of the noise strength can be estimated as in the proof of lemma 5.