Quantum computation and quantum error correction: the theoretical minimum

Abstract

These notes introduce quantum computation and quantum error correction, emphasising the importance of stabilisers and the mathematical foundations in basic Lie theory. We begin by using the double cover map $\mathrm{SU}_2 \rightarrow \mathrm{SO}_3(\mathbb{R})$ to illustrate the distinction between states and measurements for a single qubit. We then discuss entanglement and CNOT gates, the Deutsch--Jozsa Problem, and finally quantum error correction, using the Steane $[[7,1,3]]$-code as the main example. The necessary background physics of unitary evolution and Born rule measurements is developed as needed. The circuit model is used throughout.

Figures (4)

• Figure 1: The Bloch sphere showing the $Z$-basis $\left|0\right>$, $\left|1\right>$, the $X$-basis $\left|+\right>$, $\left|-\right>$ and the normalized eigenvectors of $Y = iXZ$.
• Figure 2: Final outputs of circuit for $U_f$ where $f : \mathbb{F}_2^n \rightarrow \mathbb{F}_2$ is a Boolean function: (1) graph circuit; (2) result moved to phase; (3) with $H^{\otimes n}$, computing $\sum_{w \in \mathbb{F}_2^n} c_f(w) \left|w\right> \left|-\right>$.
• Figure 3: One possible configuration of the complexity theory landscape. PP is (although not by the usual definition) the class of decision problems solvable in polynomial time with error probability for either answer strictly less than ${\frac{1}{2}}$. For example 3SAT is in PP: guess an assignment of the $n$ variables and if it works, reports SAT; otherwise report UNSAT with probability ${\frac{1}{2}} + {\frac{1}{2^{n+1}}}$. The decision problem MAJ-3SAT asks 'do the majority of assignments satisfy the 3SAT clauses': it is unlikely to have a polynomial time certificate. Decision problems based on sampling DFT coefficients, such as Deutsch--Jozsa and Simon's Problem are in $\mathsf{BQP}$, and (see the subsection on oracles below) may not in $\mathsf{P}$. It is known that $\mathsf{P} \subseteq \mathsf{NP} \subseteq \mathsf{PP} \subseteq \mathsf{PSPACE} \subseteq \mathsf{EXPTIME}$ and that at least one of these containments is proper.
• Figure 4: Preparation circuit for zero logical $\left|0\right>_L$ in the Steane code.

Theorems & Definitions (16)

• Remark 1.1
• Definition 1.2: $Z$-basis measurement
• Remark 2.3
• Definition 3.1: $Z$-basis measurement of all qubits
• Lemma 3.2: Transverse Hadamard
• proof
• Definition 3.3: Controlled gates
• Theorem 3.6: Quantum Discrete Fourier Transform in $\mathbb{F}_2^n$
• proof
• Definition 4.1