Draft:Magic state

In quantum computing, a magic state is a quantum state that, when combined with Clifford operations, enables universal quantum computing. Introduced by Sergey Bravyi and Alexei Kitaev in 2004, magic states overcome the limitations of the Gottesman–Knill theorem, which states that quantum circuits composed solely of Clifford gates and stabilizer state inputs can be efficiently simulated on a classical computer. Magic states, being non-stabilizer states, allow for non-Clifford operations, thus enabling universal quantum computation.

Quantification of magic

The degree of "magic" in a quantum state, often referred to as non-stabilizerness, is quantified using measures such as stabilizer Rényi entropies (SREs). Introduced as a family of magic monotones, SREs are theoretically robust and computationally accessible. For an $n$-qudit system, the SRE is defined as:

${\mathcal {M}}_{\alpha }(|\psi \rangle )\equiv {\frac {1}{1-\alpha }}\log \left[\sum _{{\hat {\mathcal {O}}}\in {\mathcal {W}}(n,d)}{\frac {|\langle \psi |{\hat {\mathcal {O}}}|\psi \rangle |^{2\alpha }}{d^{n}}}\right],$

where $d$ is the dimension of each qudit, ${\mathcal {W}}(n,d)$ is the Weyl–Heisenberg group modulo global phase, and $\log$ denotes the base-2 logarithm. For $\alpha \geq 2$, ${\mathcal {M}}_{\alpha }$ satisfies the properties of a resource monotone, meaning it does not increase under stabilizer operations (e.g., Clifford unitaries and projective Pauli measurements) for pure states.

Maximally magic states

Maximally magic states achieve the upper bound of ${\mathcal {M}}_{\alpha }$. For a $D$-dimensional Hilbert space, the bound is:

${\mathcal {M}}_{\alpha }(|\psi \rangle )\leq {\frac {1}{1-\alpha }}\log {\frac {1+(d-1)(d+1)^{1-\alpha }}{d}},$

achieved only by Weyl–Heisenberg covariant SIC-POVM (WH–SIC) states. The existence of WH–SICs in all finite dimensions remains an open problem, linked to Hilbert's twelfth problem in algebraic number theory. In composite n-qubit systems $(\mathbb {C} ^{2})^{\otimes n}$, WH–SICs exist only for $n=3$.

For two-qubit systems, the bound is achieved by Weyl–Heisenberg mutually unbiased bases (WH–MUB) states:

${\mathcal {M}}_{\alpha }(|\psi \rangle )\leq {\frac {1}{1-\alpha }}\log {\frac {1+(D-1)(D+\Delta )^{1-\alpha }}{D}}{\Big |}_{\Delta =0},$

where $D$ is the system’s dimension.

Single-qubit systems

For a single qubit $D=2$, analysis on the Bloch sphere identifies eight physically distinct maximally magic states, forming two WH–SICs. An example pair is:

$|M_{1}\rangle ={\frac {1}{\mathcal {N}}}{\begin{pmatrix}{\sqrt {3}}+2-i\\-1+i{\sqrt {3}}\end{pmatrix}},\quad |M_{2}\rangle ={\frac {1}{\mathcal {N}}}{\begin{pmatrix}{\sqrt {2}}+{\sqrt {6}}\\(1-i){\sqrt {2}}\end{pmatrix}},\quad {\mathcal {N}}={\frac {1}{2{\sqrt {3+{\sqrt {3}}}}}}.$

Two-qubit systems

In two-qubit systems $D=2^{2}$, 480 physically distinct maximally magic states were identified via numerical search on 14 April 2025 by Qiaofeng Liu, Ian Low, and Zhewei Yin, compared to 60 stabilizer states. On 16 June 2025, M. Ohta and K. Sakurai showed that these states, along with the 60 stabilizer states, correspond to the second-shortest vectors of the E8 lattice. Example states include:

$|M_{1}\rangle ={\frac {1}{2}}{\begin{pmatrix}0\\i\\-i\\1+i\end{pmatrix}},\quad |M_{2}\rangle ={\frac {1}{4}}{\begin{pmatrix}1+i\\1+i\\1-i\\3+i\end{pmatrix}}.$

Three-qubit systems

For three-qubit systems , 15,360 maximally magic states were identified on 16 June 2025 by M. Ohta and K. Sakurai, corresponding to the second-shortest vectors of the 16-dimensional Barnes–Wall lattice. An example WH–SIC pair is:

$|M_{1}\rangle ={\frac {1}{2{\sqrt {3}}}}(|000\rangle +|001\rangle +|010\rangle +|011\rangle +|100\rangle +(1-2i)|101\rangle -|110\rangle +|111\rangle ),$

$|M_{2}\rangle ={\frac {1}{\sqrt {6}}}((1-i)|010\rangle +|100\rangle -i|101\rangle +|110\rangle +|111\rangle ).$

Single-qutrit systems

For a single qutrit , 45 maximally magic states (WH–SICs) were derived from the second-shortest vectors of the E6 lattice on 16 June 2025. Examples include:

$|M_{1}\rangle ={\frac {1}{\sqrt {2}}}(|1\rangle +|2\rangle ),\quad |M_{2}\rangle ={\frac {1}{\sqrt {2}}}(\omega |2\rangle -|3\rangle ),$

where $\omega =e^{2\pi i/3}$

Applications

Magic states are critical for fault-tolerant quantum computing, particularly through magic state distillation, a process that concentrates magic from noisy states into high-fidelity states. This enables the implementation of non-Clifford gates, such as the T gate, essential for universal quantum computation.

See also

• Magic state distillation
• Stabilizer code
• Clifford group
• SIC-POVM
• Root system

External links

• Universal quantum computation with ideal Clifford gates and noisy ancillas – Original paper by Bravyi and Kitaev
• Stabilizer Rényi entropy – Introduction to SREs by Leone et al.