Consideraciones acerca de la carrera de Licenciatura en Tecnología de

Most quantum error correcting codes and fault tolerant circuits are designed without assuming any structure in the noise. For storing information, it is common to model the e↵ects of noise as a depolarizing noise channel

Ndep(⇢) = (1 p)⇢+p I/2, (3.1)

on each qubit, which replaces each qubit with the totally mixed state with some probabilityp. For error-correcting circuits, another useful class of error models are stochastic error models, wherer

locations are specified to be faulty, with probability_pr _{and the resulting error at each location}

can be chosen arbitrarily— even adversarially— from some specified set, usually the Pauli errors {X, Y, Z_}.

In the absence of more information about the structure of the noise, it is reasonable to take the attitude of assuming as little about the noise as possible, and these models are well justified. However, in practice, any physical implementation of a quantum computer will have additional noise structure, and it will be extremely beneficial to take into account as much of this structure as possible when designing and choosing codes and fault-tolerant gadgets. Details of the noise structure can be observed by the techniques of quantum process tomography [64–67].

Often knowledge about additional noise structure can make it harder to use error correction and fault tolerance e↵ectively. For instance, the presence of faults which are correlated in time or space can reduce the e↵ective distance of an error-correcting code, necessitating a larger code to achieve the same level of error correction. In other cases, this additional structure can lead to improvements in the e↵ectiveness of error correction, as one can tailor the choice of code and gadget to the noise situation at hand.

One common such instance is where noise can be modeled by an amplitude damping channel Ndamp(⇢) =E0⇢E0†+E1⇢E1†, (3.2)

with E0= 0 B B B @ 1 0 0 p1 1 C C C A E1= 0 B B B @ 0 p 0 0 1 C C C A, (3.3)

which describes an energy dissipating process, where there is a probability of relaxing from the excited state |1i to the ground state |0i. Under this noise model, it has been shown that an approximate recovery map can be easily constructed, allowing a simpler error correction procedure which compares favorably to other methods using exhaustive search [68]. Additionally, one can use “refrigeration” techniques to compute for an exponentially long time in the number of qubits, without an external supply of fresh ancillas [69]. Essentially this is because the amplitude damping channel itself provides the refrigeration capability to produce fresh qubits.

or simply “biased noise.” In this case, the probability "of a dephasing error Z, which causes an unexpected phase di↵erence between the_|0_iand_|1_istates, is larger than the probability"0 _{of other} errors like the bit-flip errorX, which flips the states_|0_iand _|1_i to each other. The ratio of these two rates will be called the bias :

="/"0. (3.4)

More precisely, when we consider faults that happens in circuits, we will consider a local stochas- tic biased noise model, which distinguishes between two groups of faults. The first group, dephasing faults or diagonal faults, can be modeled by inserting after the gate a trace-preserving completely positive map, whose Kraus operators are all diagonal in the computational basis. Because our fault-tolerant circuits consist of Cli↵ord gates and measurements, we can always assume that the most harmful such fault would produce a Pauli Z error (or some product of Pauli Z errors on a multi-qubit gate) after the gate. These faults will be associated with the probability ". The second group of faults, which we will call general or non-diagonal faults, are associated with the smaller probability"0_{. They may consist of any trace-preserving completely positive map, with no} restriction on the form of the Kraus operators. The most harmful of these faults will be something that produces a Pauli error. The argument also applies to more general non-diagonal faults—by expanding each Kraus operator in terms of Pauli operators, and noting that all syndrome bits are measured in theX basis, we can bound the probability for each syndrome measurement outcome in terms of" and"0 _{as above. We will assume the local stochastic condition, which says that the} sum of the probabilities for all fault paths withrdephasing faults andsgeneral faults at specified circuit locations obeys

P "r("0)s. (3.5) In our fault analysis, we will assume that, once the locations of the faults have been specified, the fault can be chosen adversarially to be most likely to cause an overall failure.

It is quite easy to envision situations in which there is a substantial dephasing noise bias when storing information for some period. In particular, it is very common to identify the computational basis{|0i,|1i}with the low-lying energy eigenstates of some physical system. In that case, there is an energetic penalty associated with flipping the state from_|0_ito _|1_i— namely, the energy gap between the two states. For thermal noise with temperatureT, the probability of a bit flip will be suppressed by a factor likee /kT_{. By contrast, dephasing errors correspond to the measuring of}

the qubit by its environment in the computational basis, which we can expect to be fairly common.