46 Def 14: Gross Code
The Gross code is a specific \([[144, 12, 12]]\) Bivariate Bicycle code, so named because a “gross” is a dozen dozens (\(144 = 12 \times 12\)).
Parameters: \(\ell = 12\), \(m = 6\), giving \(n = 2 \cdot 12 \cdot 6 = 144\) qubits.
Polynomials:
Logical operators: A convenient basis of logical operators uses the polynomials:
Then for any monomials \(\alpha , \beta \in M\):
\(\bar{X}_\alpha = X(\alpha f, 0)\) are \(X\)-type logical operators of weight 12,
\(\bar{X}'_\beta = X(\beta g, \beta h)\) are \(X\)-type logical operators,
\(\bar{Z}_\beta = Z(\beta h^T, \beta g^T)\) are \(Z\)-type logical operators,
\(\bar{Z}'_\alpha = Z(0, \alpha f^T)\) are \(Z\)-type logical operators.
The parameter \(\ell = 12\) for the Gross code.
The parameter \(m = 6\) for the Gross code.
The monomial set for the Gross code, \(\mathrm{GM} = M(\ell , m) = M(12, 6)\).
The group algebra for the Gross code, \(\mathrm{GGroupAlg} = \mathrm{GroupAlg}(12, 6)\).
Shorthand for monomials \(x^a y^b\) in the Gross code group algebra:
where \(a \in \mathbb {Z}/\ell \mathbb {Z}\) and \(b \in \mathbb {Z}/m\mathbb {Z}\).
The polynomial \(A\) for the Gross code:
The polynomial \(B\) for the Gross code:
The Gross code is a \([[144, 12, 12]]\) Bivariate Bicycle code with parameters \(\ell = 12\), \(m = 6\), and polynomials \(A = x^3 + y^2 + y\), \(B = y^3 + x^2 + x\).
The number of physical qubits in the Gross code is \(144\):
Unfolding the definition of \(\mathrm{numPhysicalQubits}\), we have \(2 \cdot \ell \cdot m = 2 \cdot 12 \cdot 6 = 144\), which follows by numerical computation.
The number of left qubits in the Gross code is \(72\):
Unfolding the definition of \(\mathrm{numLeftQubits}\), we have \(\ell \cdot m = 12 \cdot 6 = 72\), which follows by numerical computation.
The number of right qubits in the Gross code is \(72\):
Unfolding the definition of \(\mathrm{numRightQubits}\), we have \(\ell \cdot m = 12 \cdot 6 = 72\), which follows by numerical computation.
The number of \(X\) checks in the Gross code is \(72\):
Unfolding the definition of \(\mathrm{numXChecks}\), we have \(\ell \cdot m = 12 \cdot 6 = 72\), which follows by numerical computation.
The number of \(Z\) checks in the Gross code is \(72\):
Unfolding the definition of \(\mathrm{numZChecks}\), we have \(\ell \cdot m = 12 \cdot 6 = 72\), which follows by numerical computation.
We have \(n = 2 \cdot \ell \cdot m = 2 \cdot 12 \cdot 6 = 144\).
This follows by numerical computation: \(2 \times 12 \times 6 = 144\).
The polynomial \(f\) for \(X\)-type logical operators:
The polynomial \(g\) for logical operators:
The polynomial \(h\) for logical operators:
The multiplication of a monomial \(\alpha \) by a group algebra element \(p\) (left multiplication). The map \(\alpha \cdot p\) shifts all monomials in \(p\) by \(\alpha \), defined as \(\mathrm{mapDomain}(\cdot + \alpha , p)\).
For a monomial \(\alpha \in M\), the \(X\)-type logical operator \(\bar{X}_\alpha = X(\alpha f, 0)\), which has weight \(12\).
For a monomial \(\beta \in M\), the \(X\)-type logical operator \(\bar{X}'_\beta = X(\beta g, \beta h)\).
For a monomial \(\beta \in M\), the \(Z\)-type logical operator \(\bar{Z}_\beta = Z(\beta h^T, \beta g^T)\).
For a monomial \(\alpha \in M\), the \(Z\)-type logical operator \(\bar{Z}'_\alpha = Z(0, \alpha f^T)\).
For any monomial \(\alpha \), the \(\bar{Z}'_\alpha \) operator has zero left polynomial and the \(\bar{X}_\alpha \) operator has zero right polynomial:
We prove each conjunct separately. Both hold by reflexivity (definitional equality).
For any monomial \(\alpha \), we have:
This holds by the tuple \(\langle \mathrm{rfl}, \mathrm{rfl}, \mathrm{rfl}, \mathrm{rfl} \rangle \); each component is definitionally equal.
For any monomial \(\beta \), we have:
This holds by the tuple \(\langle \mathrm{rfl}, \mathrm{rfl}, \mathrm{rfl}, \mathrm{rfl} \rangle \); each component is definitionally equal.
A structure recording the code parameters \([[n, k, d]]\) of a quantum error-correcting code, where \(n\) is the number of physical qubits, \(k\) is the number of logical qubits, and \(d\) is the code distance.
The claimed parameters of the Gross code: \([[144, 12, 12]]\), i.e., \(n = 144\), \(k = 12\), \(d = 12\).
The number of physical qubits satisfies \(n = 12 \times 12 = 144\), i.e., a gross is a dozen dozens.
By numerical computation: \(12 \times 12 = 144\).
The number of physical qubits \(\mathrm{numPhysicalQubits}(\ell , m)\) equals the code parameter \(n\):
By simplification using the previously established result that \(\mathrm{numPhysicalQubits}(12, 6) = 144\).