## The Theory of Fault-Tolerance Quantum Computing

10/05/2021

Fault-tolerant error-corrected qubits are expected to have fewer defects. The problem with these qubits is that occasionally a qubit fails irrecoverably, which requires its destruction and thus prevents its correction. We have developed a technique that can preserve qubit errors under the worst-case scenario.

Introduction. The theory of fault-tolerant quantum computation. The construction of two-dimensional quantum circuits, including two-qubit gates and the implementation of unitary operations. Examples of fault-tolerant quantum computation. Some recent experimental results on fault-tolerant quantum computation. Conclusions.

The theory of fault-tolerant quantum computation has recently turned out to be the focus of intensive research [1]. Various types of quantum computation have been studied, and two-dimensional quantum circuits have been constructed for various types of quantum computation. Such circuits are based on the idea of error correction and are useful for fault-tolerant quantum computation. Two types of computation are currently being investigated. In the first type, one performs unitary computations by using an approximate unitary computation, and in the second type, one performs unitary computations with a classical computer [2].

In this paper, we review some of these results. We also examine some open problems.

In the first type of quantum computation, one performs unitary computations by using a single-qubit phase computation by a unitary quantum gate. For example, two-qubit gates with a superconducting circuit have been constructed by using the single-qubit phase computation in [3]. Note that such a unitary computation is also known as a two-qubit gate. It is also known as a controlled gate in quantum computation since the single-qubit gate is controlled by a single control qubit by applying the gate. In addition, a unitary gate is a class of unitary computations. Recently, the two-qubit gate is employed in fault-tolerant computation [4, 5] based on the quantum error-correcting code. Since a unitary computation is a unitary operation, the two-qubit gate is fault-tolerant if and only if it is a two-qubit gate. However, in fault-tolerant quantum computation, fault-like errors can be introduced through some additional errors. This is an essential difference from fault-tolerant classical computation.

In the second type of quantum computation, which is based on a classical computer, one performs unitary computations by using a classical computation.

## Implementation of the [[4, 2, 2]] code on a fully connected quantum computer.

information.

quantum algorithm for factoring was implemented.

code is used to implement the Shor algorithm.

demonstrate the quantum speedup.

applied in many areas of science.

the quantum speedup.

quantum error correction.

code is very competitive in a classical computer.

than a quantum computer that has no quantum error correction.

### Robust single-bit operations in the [[4, 2, 2]] code

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Robustly implementing single-bit operations in the [[4, 2, 2]] data type is difficult because this type of operation requires the use of a carry flag. We discuss the implementation of these operations in [[4, 2, 2]], because this implementation is of interest to computer hardware engineers. The paper discusses the implementation of the bit-shift and multiplies/divides operators of the [[4, 2, 2]] type in terms of some new hardware resources, such as the Carry Flag. Two variations are discussed: the use of a single 1-bit accumulator, and the use of an 8-bit accumulator to store a carry-flag. We also consider how to implement multiply-accumulate using the Carry Flag.

The remainder of the paper discusses the implementation of [[4, 2, 2]].

It is important to note that the article uses the word carry to refer to a carry flag, a carry-in condition, or “the condition that the carry-flag is set. ” In that context, the term “carry” means the condition that a carry flag is set.

The double-width bit-shift is described in Algorithm $alg:bit_shift$. We use the notation: $C$, $W$, $b$, $b_C$, and $b_W$ to represent $C$, the carry flag, an integer of width 2, the number of bits that are shifted, and the bit-shift operation, respectively. $W = \mathcal{S}_{b-d}$ can be calculated by: $\mathcal{S}_{b-d}$ = $0$ or $1$, according as $b = W + 1$ or $W + 1$ (see Algorithm $alg:bit_shift$). The carry flag $C$ is stored on $b_W$.

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the audience.

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being executed. It can be a CPU on your computer, or an integrated PDA.

It can be a RAM chip on your computer, or an entire hard disk.

, for use by the program being executed.