\(
\newcommand{\ket}[1]{\vert#1\rangle}
\newcommand{\bra}[1]{\langle#1\vert}
\newcommand{\bigket}[1]{\bigl\vert#1\bigr\rangle}
\newcommand{\bigbra}[1]{\bigl\langle#1\bigr\vert}
\newcommand{\class}[1]{\mathrm{#1}}
\newcommand{\problem}[1]{\mathrm{#1}}
\renewcommand{\natural}{\mathbb{N}}
\newcommand{\real}{\mathbb{R}}
\newcommand{\complex}{\mathbb{C}}
\newcommand{\ip}[2]{\left\langle#1, #2 \right\rangle}
\newcommand{\Tr}{\mathop{\mathrm{Tr}}\nolimits}
\newcommand{\tr}{\mathop{\mathrm{tr}}\nolimits}
\newcommand{\abs}[1]{\left\lvert#1 \right\rvert}
\newcommand{\norm}[1]{\left\lVert#1 \right\rVert}
\newcommand{\floor}[1]{\left\lfloor#1 \right\rfloor}
\newcommand{\X}{\mathcal{X}}
\newcommand{\Y}{\mathcal{Y}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\E}{\mathop{\mathbb{E}}}
\newcommand{\Var}{\mathop{\mathrm{Var}}}
\newcommand{\dif}{\mathrm{d}}
\newcommand{\eps}{\epsilon}
\newcommand{\sign}{\mathrm{sign}}
\newcommand{\poly}{\mathrm{poly}}
\newcommand{\polylog}{\mathrm{polylog}}
\newcommand{\negl}{\mathrm{negl}}
\newcommand{\church}[1]{\overline{#1}}
\newcommand{\defeq}{\stackrel{\text{def}}{=}}
\newcommand{\pair}[2]{\langle#1, #2\rangle}
\newcommand{\tuple}[1]{\langle#1\rangle}
\newcommand{\red}{\rightarrow}
\newcommand{\reds}{\twoheadrightarrow}
\newcommand{\betared}{\rightarrow_{\beta}}
\newcommand{\betareds}{\twoheadrightarrow_{\beta}}
\newcommand{\betaeq}{=_{\beta}}
\newcommand{\betaetared}{\rightarrow_{\beta\eta}}
\newcommand{\betaetareds}{\twoheadrightarrow_{\beta\eta}}
\newcommand{\parared}{\rightarrow_{\|}}
\newcommand{\parareds}{\twoheadrightarrow_{\|}}
\newcommand{\desc}[1]{\langle#1 \rangle}
\newcommand{\adv}{\mathcal{A}}
\newcommand{\dis}{\mathcal{D}}
\newcommand{\labelsty}[1]{\mathrm{#1}}
\newcommand{\Enc}{\labelsty{Enc}}
\newcommand{\Dec}{\labelsty{Dec}}
\newcommand{\plain}{\labelsty{p}}
\newcommand{\cipher}{\labelsty{c}}
\newcommand{\PRG}{\labelsty{PRG}}
\newcommand{\Commit}{\labelsty{Com}}
\newcommand{\oracle}{\mathcal{O}}
\newcommand{\Sim}{\labelsty{Sim}}
\newcommand{\comp}{\labelsty{c}}
\newcommand{\view}{\labelsty{view}}
\newcommand\TIME{\class{TIME}}
\newcommand\SPACE{\class{SPACE}}
\newcommand\SIZE{\class{SIZE}}
\newcommand\NTIME{\class{NTIME}}
\newcommand\NSPACE{\class{NSPACE}}
\newcommand\BQP{\class{BQP}}
\newcommand\BPP{\class{BPP}}
\newcommand\QMA{\class{QMA}}
\renewcommand\P{\class{P}}
\newcommand\Ppoly{\class{P{/}poly}}
\newcommand\NP{\class{NP}}
\newcommand\NPC{\class{NP}\text{-complete}}
\newcommand\coNP{\class{coNP}}
\newcommand\MA{\class{MA}}
\newcommand\AM{\class{AM}}
\newcommand\QCMA{\class{QCMA}}
\newcommand\EXP{\class{EXP}}
\newcommand\NEXP{\class{NEXP}}
\newcommand\PP{\class{PP}}
\newcommand\GapP{\class{GapP}}
\newcommand\IP{\class{IP}}
\newcommand\QIP{\class{QIP}}
\newcommand\PSPACE{\class{PSPACE}}
\newcommand\NPSPACE{\class{NPSPACE}}
\newcommand\MIP{\class{MIP}}
\newcommand\RE{\class{RE}}
\newcommand\QMAM{\class{QMAM}}
\newcommand\PH{\class{PH}}
\renewcommand\L{\class{L}}
\newcommand\NL{\class{NL}}
\newcommand\coNL{\class{coNL}}
\newcommand\NLC{\class{NL}\text{-complete}}
\newcommand\PPAD{\class{PPAD}}
\newcommand\SharpP{\#\class{P}}
\newcommand\HVPZK{\class{HVPZK}}
\newcommand\HVSZK{\class{HVSZK}}
\newcommand\PZK{\class{PZK}}
\newcommand\SZK{\class{SZK}}
\newcommand\CZK{\class{CZK}}
\newcommand\val{\mathrm{val}}
\newcommand\vale{\mathrm{val}^*}
\newcommand\perm{\mathop{\mathrm{perm}}\nolimits}
\newcommand\pathfunc{\mathrm{path}}
\newcommand\SELF{\problem{SELF}}
\newcommand\HALT{\problem{HALT}}
\newcommand\MINTM{\problem{MIN}_\text{TM}}
\newcommand\ACCTM{\problem{ACC}_\text{TM}}
\newcommand\TQBF{\problem{TQBF}}
\newcommand\PAL{\problem{PAL}}
\newcommand\PATH{\problem{PATH}}
\newcommand\LONGPATH{\problem{LONGEST\text{-}PATH}}
\newcommand\SHORTPATH{\problem{SHORTEST\text{-}PATH}}
\newcommand\HAMPATH{\problem{HAM\text{-}PATH}}
\newcommand\HAMCYCLE{\problem{HAM\text{-}CYCLE}}
\newcommand\INDSET{\problem{IND\text{-}SET}}
\newcommand\MINCUT{\problem{MIN\text{-}CUT}}
\newcommand\MAXFLOW{\problem{MAX\text{-}FLOW}}
\newcommand\MAXCUT{\problem{MAX\text{-}CUT}}
\newcommand\MAXCUTAPP{\problem{MAX\text{-}CUT\text{-}APPROX}}
\newcommand\SAT{\problem{SAT}}
\newcommand\PERM{\problem{PERMANENT}}
\newcommand\THREESAT{3\text{-}\problem{SAT}}
\newcommand\TWOSAT{2\text{-}\problem{SAT}}
\newcommand\TWOUNSAT{2\text{-}\problem{UNSAT}}
\newcommand\NAESAT{\problem{NAESAT}}
\newcommand\REACHABILITY{\problem{REACHABILITY}}
\newcommand\THREECOL{3\text{-}\problem{COLORING}}
\newcommand\CLIQUE{\problem{CLIQUE}}
\newcommand\COL{\problem{COLORING}}
\newcommand\COMPOSITE{\problem{COMPOSITE}}
\newcommand\PRIME{\problem{PRIME}}
\newcommand\TSP{\problem{TSP}}
\newcommand\VC{\problem{VERTEX\text{-}COVER}}
\newcommand\FACTOR{\problem{FACTOR}}
\newcommand\GI{\problem{GRAPH\text{-}ISO}}
\newcommand\GNI{\problem{GRAPH\text{-}NONISO}}
\newcommand\SHORTPROOF{\problem{SHORT\text{-}PROOF}}
\newcommand\UHALT{\problem{UNARY\text{-}HALT}}
\newcommand\GG{\problem{GENERAL\text{-}GEOGRAPHY}}
\newcommand\FG{\problem{FORMULA\text{-}GAME}}
\)
Background
In this project, you will explore problems related to transversal
Clifford gates and investigate quantum codes that achieve specific
transversal Clifford operations. The study of quantum codes and
transversal gates is crucial for quantum fault-tolerant computation,
which is a fundamental requirement for building reliable quantum
computers. You may refer to Chapter 10 of [NC00] and Chapter 11 of
[Got24] for foundational knowledge on this topic.
Definitions
We introduce two definitions of transversal gates:
Consider a quantum code \(\mathcal{C}
\subset \mathcal{H}^{\otimes n}\), with an encoding map \(C: \mathcal{H} \rightarrow \mathcal{H}^{\otimes
n}\) and a projector \(\Pi_{\mathcal{C}}\) onto the code space.
For any physical quantum state \(|\psi\rangle
\in \mathcal{H}^{\otimes k}\), its logical counterpart under the
code scheme \(\mathcal{C}\) is denoted
as \(|\psi\rangle_L := C^{\otimes
k}|\psi\rangle \in (\mathcal{H}^{\otimes k})^{\otimes n}\).
Let \(\mathcal{G}\) be the set of
all permitted quantum gates on physical qubits. For any quantum gate
\(G \in \mathcal{G}\), its logical
implementation under the code scheme \(\mathcal{C}\) is denoted as \(G_L\), satisfying the condition \(G_L
|\psi\rangle_L = (G |\psi\rangle)_L\) for any physical quantum
state \(|\psi\rangle\).
We define two types of transversal gates:
- Strict Transversality: A code \(\mathcal{C}\) admits a strictly transversal
logical gate \(G_L\) for \(G \in \mathcal{G}\) if \(G_L \ket{\psi}_L =G^{\otimes
n} \ket{\psi}_L\) for \(\ket{\psi}_L\) encoding any \(k\)-qubit state \(\ket{\psi}\) if \(G\) is a gate of \(k\) qubits.
- Weak Transversality: A code \(\mathcal{C}\) admits a weakly transversal
logical gate \(G_L\) for \(G \in \mathcal{G}\) if \(G_L \ket{\psi}_L =G^{\otimes
n} \ket{\psi}_L\) for \(\ket{\psi}_L\) encoding any \(k\)-qubit state \(\ket{\psi}\) if \(G\) is a gate of \(k\) qubits.
Two well-known results are that all stabilizer codes achieve strict
transversal Pauli gates, and all CSS codes achieve strict transversal
CNOT gates. Transversal operations are crucial for fault-tolerant
quantum computation as they prevent error propagation across code
blocks.
Problems
This project consists of three tasks, progressing in complexity. The
first is relatively straightforward and can be found in [Got16], while
the second is research-level and may benefit from insights in [ZCC11].
The third is significantly more challenging; solving it could lead to a
publishable result in quantum information theory, and [GG24] provides
relevant background.
- Strictly Transversal Hadamard and Phase Gates:
Identify quantum codes that achieve strict transversal Hadamard and
Phase gates. Derive necessary and sufficient conditions for their
existence.
- Weakly Transversal Hadamard and Phase Gates:
Identify quantum codes that achieve weak transversal Hadamard and Phase
gates. Provide necessary and sufficient conditions, or present them
separately if a full characterization is difficult.
- QLDPC Codes with Transversal Hadamard Gates:
Investigate whether good QLDPC codes exist that support transversal
Hadamard gates (either strict or weak). Provide explicit constructions
or proofs of existence.
References
- [NC00] Nielsen, M. A., & Chuang, I. L. (2010). Quantum
Computation and Quantum Information. Cambridge University
Press.
- [Got16] Gottesman, D. (2016). Surviving as a quantum computer in a
classical world. Textbook manuscript preprint.
- [ZCC11] Zeng, B., Cross, A., & Chuang, I. L. (2011).
Transversality versus universality for additive quantum codes. IEEE
Transactions on Information Theory, 57(9), 6272-6284.
- [GG24] Golowich, L., & Guruswami, V. (2024). Asymptotically good
quantum codes with transversal non-Clifford gates. arXiv preprint
arXiv:2408.09254.