Transversal Gates¶

Transversal gates are important in fault-tolerant quantum computing, as they allow for operations that can be performed on multiple qubits simultaneously without introducing errors that propagate through the system. These gates are particularly useful in stabilizer codes and other error-correcting codes. This tutorial will cover the basics of transversal gates and their properties, especially reproduce results in the arxiv paper arxiv-link.

In [1]:

import functools
import numpy as np

try:
    import numqi
except ImportError:
    %pip install numqi
    import numqi

np_rng = np.random.default_rng()
hf_kron = lambda *x: functools.reduce(np.kron, x) #tensor product of matrices

import functools import numpy as np try: import numqi except ImportError: %pip install numqi import numqi np_rng = np.random.default_rng() hf_kron = lambda *x: functools.reduce(np.kron, x) #tensor product of matrices

Stabilizer codes¶

Let's begin with two famous stabilizer codes: the 5-qubit code and the Steane code.

5-qubit code ((5,2,3))¶

A quantum error-correcting code (QECC) is a subspace specified by a set of logical basis states

In [2]:

code,info = numqi.qec.get_code_subspace('523')

# logical states are some special state vector in the complete Hilbert space
print('logical 0:', (code[0]*4).astype(np.int64), sep='\n')
print('logical 1:', (code[1]*4).astype(np.int64), sep='\n')

code,info = numqi.qec.get_code_subspace('523') # logical states are some special state vector in the complete Hilbert space print('logical 0:', (code[0]*4).astype(np.int64), sep='\n') print('logical 1:', (code[1]*4).astype(np.int64), sep='\n')

logical 0:
[ 1  0  0 -1  0  1 -1  0  0  1  1  0 -1  0  0 -1  0 -1  1  0  1  0  0 -1
 -1  0  0 -1  0 -1 -1  0]
logical 1:
[ 0  1  1  0  1  0  0  1  1  0  0 -1  0 -1  1  0  1  0  0  1  0 -1 -1  0
  0  1 -1  0  1  0  0 -1]

Via basis matrix multiplication, one can verify that logical X/Y/Z gate can be implemented by applying the Pauli gates on all physical qubits. Such gates are called transversal gates.

In [3]:

# logical X = X * X * X * X * X
logicalX = code.conj() @ hf_kron(*[-numqi.gate.X]*5) @ code.T
print('logical X:', logicalX, sep='\n')

# logical Y = Y * Y * Y * Y * Y
logicalY = code.conj() @ hf_kron(*[-numqi.gate.Y]*5) @ code.T
print('\nlogical Y:', logicalY, sep='\n')

# logical Z = Z * Z * Z * Z * Z
logicalZ = code.conj() @ hf_kron(*[numqi.gate.Z]*5) @ code.T
print('\nlogical Z:', logicalZ, sep='\n')

# logical X = X * X * X * X * X logicalX = code.conj() @ hf_kron(*[-numqi.gate.X]*5) @ code.T print('logical X:', logicalX, sep='\n') # logical Y = Y * Y * Y * Y * Y logicalY = code.conj() @ hf_kron(*[-numqi.gate.Y]*5) @ code.T print('\nlogical Y:', logicalY, sep='\n') # logical Z = Z * Z * Z * Z * Z logicalZ = code.conj() @ hf_kron(*[numqi.gate.Z]*5) @ code.T print('\nlogical Z:', logicalZ, sep='\n')

logical X:
[[0. 1.]
 [1. 0.]]

logical Y:
[[0.+0.j 0.-1.j]
 [0.+1.j 0.+0.j]]

logical Z:
[[ 1.  0.]
 [ 0. -1.]]

Besides transversal logical X/Y/Z gates, the 5-qubit code also has transversal logical $F=HS^\dagger$ gate (in Bloch sphere, $F$ gate is a rotation by 120 degrees around (1,1,1) axis). Although 5-qubit code has transversal logical $F=HS^\dagger$, but it does not have transversal logical $H$ and $S$ gates. To support transversal logical $H$ and $S$ gates, we need to use the Steane code which will be introduced in the next section.

In [4]:

op_physical = numqi.gate.Y @ numqi.gate.H @ numqi.gate.S.conj()
logicalF = -np.exp(1j*np.pi/4)*code.conj() @ hf_kron(*[op_physical]*5) @ code.T
print('logical F:', logicalF, sep='\n')

op_physical = numqi.gate.Y @ numqi.gate.H @ numqi.gate.S.conj() logicalF = -np.exp(1j*np.pi/4)*code.conj() @ hf_kron(*[op_physical]*5) @ code.T print('logical F:', logicalF, sep='\n')

logical F:
[[ 0.5-0.5j -0.5-0.5j]
 [ 0.5-0.5j  0.5+0.5j]]

Besides transversal gates, numqi also calculate Shor's weight enumerator of the code. Code distance can be read from the weight enumerator which is the first non-equal term in the weight enumerator with its dual. For the 5-qubit code, the distance is 3, which means A3,B3 are different.

In [5]:

print('Shor weight enumerator (A0,A1,A2,A3,A4,A5):', info['qweA'], sep='\n')
print('\ndual of Shor weight enumerator (B0,B1,B2,B3,B4,B5):', info['qweB'], sep='\n')

print('Shor weight enumerator (A0,A1,A2,A3,A4,A5):', info['qweA'], sep='\n') print('\ndual of Shor weight enumerator (B0,B1,B2,B3,B4,B5):', info['qweB'], sep='\n')

Shor weight enumerator (A0,A1,A2,A3,A4,A5):
[ 1  0  0  0 15  0]

dual of Shor weight enumerator (B0,B1,B2,B3,B4,B5):
[ 1  0  0 30 15 18]

Steane code ((7,2,3))¶

In [6]:

code,info = numqi.qec.get_code_subspace('steane')

print('code.shape:', code.shape)
print('stabilizer:', info['stab'])
print('Shor weight enumerator:', info['qweA'])
print('dual of Shor weight enumerator:', info['qweB'])

code,info = numqi.qec.get_code_subspace('steane') print('code.shape:', code.shape) print('stabilizer:', info['stab']) print('Shor weight enumerator:', info['qweA']) print('dual of Shor weight enumerator:', info['qweB'])

code.shape: (2, 128)
stabilizer: ['IIXIXXX', 'IXIXXXI', 'XIIXIXX', 'IIZIZZZ', 'IZIZZZI', 'ZIIZIZZ']
Shor weight enumerator: [ 1  0  0  0 21  0 42  0]
dual of Shor weight enumerator: [  1   0   0  21  21 126  42  45]

Steane code support transversal logical $H$ and $S$ gates, as demonstrated below.

TASK: given these two transversal logical gates, can you find the implementation of transversal logical gate $F=HS^\dagger$?

In [7]:

# logical H = H * H * H * H * H * H * H
logicalH = code.conj() @ hf_kron(*[numqi.gate.H]*7) @ code.T
print('logical H:', logicalH, sep='\n')

# logical S = (S * S * S * S * S * S * S)^\dagger
logicalS = code.conj() @ hf_kron(*[numqi.gate.S.conj()]*7) @ code.T
print('\nlogical S:', logicalS, sep='\n')

# logical H = H * H * H * H * H * H * H logicalH = code.conj() @ hf_kron(*[numqi.gate.H]*7) @ code.T print('logical H:', logicalH, sep='\n') # logical S = (S * S * S * S * S * S * S)^\dagger logicalS = code.conj() @ hf_kron(*[numqi.gate.S.conj()]*7) @ code.T print('\nlogical S:', logicalS, sep='\n')

logical H:
[[ 0.70710678  0.70710678]
 [ 0.70710678 -0.70710678]]

logical S:
[[1.+0.j 0.+0.j]
 [0.+0.j 0.+1.j]]

Actually, transversal logical gates make a group: given two transversal logical gates, then their product is also a transversal logical gate.

Furthermore, Eastin-Knill theorem states that any transversal group of nontrivial (distance >1) QECC is a finite subgroup of SU(K) where K is the dimension of the logical subspace.

Finite subgroup of SU(2) has been classified as follows: cyclic groups $C_{2m}$, binary dihedral groups $\mathrm{BD}_{2m}$, and the three exceptional groups: binary tetrahedral group (2T), binary octahedral group (2O), and binary icosahedral group (2I).

group	notable elements	generators	order
$C_{2m}$	$Z(2\pi/m)$	$Z(2\pi/m)$	2m
$\mathrm{BD}_{2m}$	$\hat{X},Z(2\pi/m)$	$\hat{X},Z(2\pi/m)$	4m
2T	$\hat{X},\hat{Z},F$	$\hat{X},F$	24
2O (Clifford)	$\hat{X},\hat{Z},\hat{S},\hat{H},F$	$\hat{S},\hat{H}$	48
2I	$\hat{X},\hat{Z},F,\Phi$	$\hat{X},\hat{Z}\Phi$	120

According to classification, the transversal group of the 5-qubit code is 2T, and the transversal group of the Steane code is 2O which is also isomorphic to 1-qubit Clifford group.

Transversal group of stabilizer code is quite limited, below we demonstrate that transversal group of non-stabilizer code is much richer, which is the main result of our paper arxiv-link.

Non-stabilizer code ((6,2,3))¶

C10¶

In [8]:

code,info = numqi.qec.q623.get_C10(return_info=True)

print('code.shape:', code.shape)
print('Shor weight enumerator:', info['qweA'])
print('dual of Shor weight enumerator:', info['qweB'])

code,info = numqi.qec.q623.get_C10(return_info=True) print('code.shape:', code.shape) print('Shor weight enumerator:', info['qweA']) print('dual of Shor weight enumerator:', info['qweB'])

code.shape: (2, 64)
Shor weight enumerator: [ 1.    0.    0.84  0.   11.64 15.36  3.16]
dual of Shor weight enumerator: [ 1.    0.    0.84 23.36 36.6  39.36 26.84]

info['su2'] stores the SU(2) gates applied to each qubits that implement a Z-rotation logical gate $Z(2\pi/5)$.

In [9]:

print('physical gates for each qubit:')
for i0,op in enumerate(info['su2']):
    print(f'qubit {i0}:', op, sep='\n')

op_logical = code.conj() @ hf_kron(*info['su2']) @ code.T
print('\nlogical Z(2pi/5):', op_logical, sep='\n')

print('physical gates for each qubit:') for i0,op in enumerate(info['su2']): print(f'qubit {i0}:', op, sep='\n') op_logical = code.conj() @ hf_kron(*info['su2']) @ code.T print('\nlogical Z(2pi/5):', op_logical, sep='\n')

physical gates for each qubit:
qubit 0:
[[0.80901699-0.58778525j 0.        +0.j        ]
 [0.        +0.j         0.80901699+0.58778525j]]
qubit 1:
[[0.80901699-0.58778525j 0.        +0.j        ]
 [0.        +0.j         0.80901699+0.58778525j]]
qubit 2:
[[0.80901699-0.58778525j 0.        +0.j        ]
 [0.        +0.j         0.80901699+0.58778525j]]
qubit 3:
[[0.80901699-0.58778525j 0.        +0.j        ]
 [0.        +0.j         0.80901699+0.58778525j]]
qubit 4:
[[0.30901699-0.95105652j 0.        +0.j        ]
 [0.        +0.j         0.30901699+0.95105652j]]
qubit 5:
[[-0.30901699-0.95105652j  0.        +0.j        ]
 [ 0.        +0.j         -0.30901699+0.95105652j]]

logical Z(2pi/5):
[[0.80901699+0.58778525j 0.        +0.j        ]
 [0.        +0.j         0.80901699-0.58778525j]]

SO(5) code¶

when vece has at least 4 nonzero entries, then the corresponding code has no transversal logical gates except trivial identity, C2 group.

In [10]:

tmp0 = np_rng.normal(size=4)
vece = tmp0 / np.linalg.norm(tmp0)
code = numqi.qec.q623.get_SO5_code_with_transversal_gate(vece) #no trasversal gate

tmp0 = np_rng.normal(size=4) vece = tmp0 / np.linalg.norm(tmp0) code = numqi.qec.q623.get_SO5_code_with_transversal_gate(vece) #no trasversal gate

when 5-dimensional vece has at most 3 nonzero entries, then transversal group is C4.

In [11]:

tmp0 = np_rng.normal(size=3)
vece = tmp0 / np.linalg.norm(tmp0)
code,info = numqi.qec.q623.get_SO5_code_with_transversal_gate(vece) #no trasversal gate

print("physical gates' shape:", info['su2'].shape)
logicalZ = code.conj() @ hf_kron(*info['su2']) @ code.T
print('logical Z:', logicalZ, sep='\n')

tmp0 = np_rng.normal(size=3) vece = tmp0 / np.linalg.norm(tmp0) code,info = numqi.qec.q623.get_SO5_code_with_transversal_gate(vece) #no trasversal gate print("physical gates' shape:", info['su2'].shape) logicalZ = code.conj() @ hf_kron(*info['su2']) @ code.T print('logical Z:', logicalZ, sep='\n')

physical gates' shape: (6, 2, 2)
logical Z:
[[ 1.00000000e+00+0.00000000e+00j  2.48711257e-18-4.61769199e-20j]
 [-2.48711257e-18-4.61769199e-20j -1.00000000e+00+0.00000000e+00j]]

when vece has at most 2 nonzero entries, then transversal group is BD4.

In [12]:

tmp0 = np_rng.normal(size=2)
vece = tmp0 / np.linalg.norm(tmp0)
code,info = numqi.qec.q623.get_SO5_code_with_transversal_gate(vece) #no trasversal gate

logicalZ = code.conj() @ hf_kron(*info['su2Z']) @ code.T
print('logical Z:', np.around(logicalZ,10), sep='\n')

logicalX = code.conj() @ hf_kron(*info['su2X']) @ code.T
print('logical X:', np.around(logicalX,10), sep='\n')

tmp0 = np_rng.normal(size=2) vece = tmp0 / np.linalg.norm(tmp0) code,info = numqi.qec.q623.get_SO5_code_with_transversal_gate(vece) #no trasversal gate logicalZ = code.conj() @ hf_kron(*info['su2Z']) @ code.T print('logical Z:', np.around(logicalZ,10), sep='\n') logicalX = code.conj() @ hf_kron(*info['su2X']) @ code.T print('logical X:', np.around(logicalX,10), sep='\n')

logical Z:
[[ 1.+0.j  0.+0.j]
 [ 0.+0.j -1.+0.j]]
logical X:
[[ 0.+0.j  1.-0.j]
 [ 1.-0.j -0.-0.j]]

((6,2,3)) from ((5,2,3)) stabilizer code¶

see appendix of arxiv-link for details.

Non-stabilizer code ((7,2,3))¶

2T, Cyclic code¶

arxiv-link Characterizing Quantum Codes via the Coefficients in Knill-Laflamme Conditions

Parametrized with the signature norm $\lambda^*\in[0,\sqrt{7}]$, when $\lambda^*=0$, it becomes the Steane code, and when $\lambda^*=\sqrt{7}$, it becomes a permutational-invariant code.

In [13]:

# 0<lambda<sqrt(7), Cyclic code, 2T
code,info = numqi.qec.q723.get_cyclic_code(lambda2=np_rng.uniform(0,7), sign='++', return_info=True)

print('code.shape:', code.shape)
print('Shor weight enumerator:', info['qweA'])
print('dual of Shor weight enumerator:', info['qweB'])

# 0

code.shape: (2, 128)
Shor weight enumerator: [ 1.          0.          5.05043361  0.         10.89913277  0.
 47.05043361  0.        ]
dual of Shor weight enumerator: [ 1.          0.          5.05043361 36.15130084 10.89913277 95.69739831
 47.05043361 60.15130084]

In [14]:

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T
print('\nlogical X:', logicalX, sep='\n')

logicalF = code.conj() @ hf_kron(*[numqi.gate.H @ numqi.gate.S.conj()]*7) @ code.T
print('\nlogical F:', logicalF, sep='\n')

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T print('\nlogical X:', logicalX, sep='\n') logicalF = code.conj() @ hf_kron(*[numqi.gate.H @ numqi.gate.S.conj()]*7) @ code.T print('\nlogical F:', logicalF, sep='\n')

logical X:
[[0. 1.]
 [1. 0.]]

logical F:
[[0.70710678+0.j         0.        +0.70710678j]
 [0.70710678+0.j         0.        -0.70710678j]]

When $\lambda^*=0$, extra transversal logical gates are $H$ and $S$, which is the same as Steane code.

In [15]:

code,info = numqi.qec.q723.get_cyclic_code(lambda2=0, sign='++', return_info=True)

logicalH = code.conj() @ hf_kron(*[numqi.gate.H]*7) @ code.T
print('logical H:', logicalH, sep='\n')

logicalS = code.conj() @ hf_kron(*[numqi.gate.S.conj()]*7) @ code.T
print('\nlogical S:', logicalS, sep='\n')

code,info = numqi.qec.q723.get_cyclic_code(lambda2=0, sign='++', return_info=True) logicalH = code.conj() @ hf_kron(*[numqi.gate.H]*7) @ code.T print('logical H:', logicalH, sep='\n') logicalS = code.conj() @ hf_kron(*[numqi.gate.S.conj()]*7) @ code.T print('\nlogical S:', logicalS, sep='\n')

logical H:
[[ 0.70710678  0.70710678]
 [ 0.70710678 -0.70710678]]

logical S:
[[1.+0.j 0.+0.j]
 [0.+0.j 0.+1.j]]

When $\lambda^*=\sqrt{7}$, extra transversal logical gate is $\Phi$.

In [16]:

code,info = numqi.qec.q723.get_cyclic_code(lambda2=7, sign='++', return_info=True)

physical_op = numqi.qec.su2_finite_subgroup_gate_dict['Phi']
logicalPhi = code.conj() @ hf_kron(*[physical_op]*7) @ code.T
print('logical Psi:', logicalPhi, sep='\n')

code,info = numqi.qec.q723.get_cyclic_code(lambda2=7, sign='++', return_info=True) physical_op = numqi.qec.su2_finite_subgroup_gate_dict['Phi'] logicalPhi = code.conj() @ hf_kron(*[physical_op]*7) @ code.T print('logical Psi:', logicalPhi, sep='\n')

logical Psi:
[[-0.30901699+8.09016994e-01j  0.5       +2.09867395e-17j]
 [-0.5       +3.12223623e-17j -0.30901699-8.09016994e-01j]]

2I, permutation-invariant code¶

In [17]:

coeff = np.zeros((2,8), dtype=np.complex128)
coeff[[0,0,1,1],[0,5,2,7]] = np.array([np.sqrt(3), np.sqrt(7)*1j, np.sqrt(7)*1j, np.sqrt(3)]) / np.sqrt(10)
code = coeff @ (numqi.dicke.get_dicke_basis(7, 2)[::-1])

qweA,qweB = numqi.qec.get_weight_enumerator(code)
print('Shor weight enumerator:', np.around(qweA,3))
print('dual of Shor weight enumerator:', np.around(qweB,3))

coeff = np.zeros((2,8), dtype=np.complex128) coeff[[0,0,1,1],[0,5,2,7]] = np.array([np.sqrt(3), np.sqrt(7)*1j, np.sqrt(7)*1j, np.sqrt(3)]) / np.sqrt(10) code = coeff @ (numqi.dicke.get_dicke_basis(7, 2)[::-1]) qweA,qweB = numqi.qec.get_weight_enumerator(code) print('Shor weight enumerator:', np.around(qweA,3)) print('dual of Shor weight enumerator:', np.around(qweB,3))

Shor weight enumerator: [ 1.  0.  7.  0.  7.  0. 49.  0.]
dual of Shor weight enumerator: [ 1.  0.  7. 42.  7. 84. 49. 66.]

In [18]:

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T
print('logical X:', logicalX, sep='\n')

logicalZ5 = code.conj() @ hf_kron(*[numqi.gate.rz(6*np.pi/5)]*7) @ code.T
print('\nlogical Z(2pi/5):', logicalZ5, sep='\n')

hfR = lambda a,b,t=1: numqi.gate.I*np.cos(t*np.pi/5) + 1j*np.sin(t*np.pi/5)/np.sqrt(5) * (a*numqi.gate.Y + b*numqi.gate.Z)
physical_op = hfR(-2,-1,3)
logicalR = code.conj() @ hf_kron(*[physical_op]*7) @ code.T #hfR(-2,1)
print('\nlogical R(-2,1):', logicalR, sep='\n')

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T print('logical X:', logicalX, sep='\n') logicalZ5 = code.conj() @ hf_kron(*[numqi.gate.rz(6*np.pi/5)]*7) @ code.T print('\nlogical Z(2pi/5):', logicalZ5, sep='\n') hfR = lambda a,b,t=1: numqi.gate.I*np.cos(t*np.pi/5) + 1j*np.sin(t*np.pi/5)/np.sqrt(5) * (a*numqi.gate.Y + b*numqi.gate.Z) physical_op = hfR(-2,-1,3) logicalR = code.conj() @ hf_kron(*[physical_op]*7) @ code.T #hfR(-2,1) print('\nlogical R(-2,1):', logicalR, sep='\n')

logical X:
[[0.+0.j 1.+0.j]
 [1.+0.j 0.+0.j]]

logical Z(2pi/5):
[[0.80901699-0.58778525j 0.        +0.j        ]
 [0.        +0.j         0.80901699+0.58778525j]]

logical R(-2,1):
[[ 0.80901699+2.62865556e-01j -0.52573111+1.23259516e-32j]
 [ 0.52573111+1.23259516e-32j  0.80901699-2.62865556e-01j]]

2I, lambda*=0¶

In [19]:

code,info = numqi.qec.q723.get_2I_lambda0(theta=np_rng.uniform(0,2*np.pi),
                phi=np_rng.uniform(0,2*np.pi), sign='+', return_info=True)

print('code.shape:', code.shape)
print('Shor weight enumerator:', info['qweA'])
print('dual of Shor weight enumerator:', info['qweB'])

code,info = numqi.qec.q723.get_2I_lambda0(theta=np_rng.uniform(0,2*np.pi), phi=np_rng.uniform(0,2*np.pi), sign='+', return_info=True) print('code.shape:', code.shape) print('Shor weight enumerator:', info['qweA']) print('dual of Shor weight enumerator:', info['qweB'])

code.shape: (2, 128)
Shor weight enumerator: [ 1.          0.          0.          0.         20.80396554  0.58810338
 41.41189662  0.19603446]
dual of Shor weight enumerator: [  1.           0.           0.          20.80396554  21.78413784
 124.82379323  42.78413784  44.80396554]

In [20]:

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T
print('logical X:', logicalX, sep='\n')

logicalZ5 = code.conj() @ hf_kron(*info['su2']) @ code.T
print('\nlogical Z(2pi/5):', logicalZ5, sep='\n')

# hfR = lambda a,b,t=1: numqi.gate.I*np.cos(t*np.pi/5) + 1j*np.sin(t*np.pi/5)/np.sqrt(5) * (a*numqi.gate.Y + b*numqi.gate.Z)
# physical_op = hfR(-2,-1,3)
logicalR = code.conj() @ hf_kron(*info['su2R']) @ code.T #hfR(-2,1)
print('\nlogical R(-2,1):', logicalR, sep='\n')

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T print('logical X:', logicalX, sep='\n') logicalZ5 = code.conj() @ hf_kron(*info['su2']) @ code.T print('\nlogical Z(2pi/5):', logicalZ5, sep='\n') # hfR = lambda a,b,t=1: numqi.gate.I*np.cos(t*np.pi/5) + 1j*np.sin(t*np.pi/5)/np.sqrt(5) * (a*numqi.gate.Y + b*numqi.gate.Z) # physical_op = hfR(-2,-1,3) logicalR = code.conj() @ hf_kron(*info['su2R']) @ code.T #hfR(-2,1) print('\nlogical R(-2,1):', logicalR, sep='\n')

logical X:
[[0.+0.j 1.+0.j]
 [1.+0.j 0.+0.j]]

logical Z(2pi/5):
[[0.80901699-0.58778525j 0.        +0.j        ]
 [0.        +0.j         0.80901699+0.58778525j]]

logical R(-2,1):
[[ 0.80901699+2.62865556e-01j -0.52573111+6.93889390e-18j]
 [ 0.52573111-6.93889390e-18j  0.80901699-2.62865556e-01j]]

2I, lambda*=0.75¶

In [21]:

code,info = numqi.qec.q723.get_2I_lambda075(np_rng.uniform(0,np.sqrt(5/16)),
                        sign=np.array([1,1,1]), return_info=True)

print('code.shape:', code.shape)
print('Shor weight enumerator:', info['qweA'])
print('dual of Shor weight enumerator:', info['qweB'])

code,info = numqi.qec.q723.get_2I_lambda075(np_rng.uniform(0,np.sqrt(5/16)), sign=np.array([1,1,1]), return_info=True) print('code.shape:', code.shape) print('Shor weight enumerator:', info['qweA']) print('dual of Shor weight enumerator:', info['qweB'])

code.shape: (2, 128)
Shor weight enumerator: [ 1.          0.          0.75        0.         18.05903865  4.32288405
 38.42711595  1.44096135]
dual of Shor weight enumerator: [  1.           0.           0.75        21.80903865  25.2638454
 112.8542319   48.5138454   45.80903865]

In [22]:

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T
print('logical X:', logicalX, sep='\n')

logicalZ5 = code.conj() @ hf_kron(*info['su2']) @ code.T
print('\nlogical Z(2pi/5):', logicalZ5, sep='\n')

# hfR = lambda a,b,t=1: numqi.gate.I*np.cos(t*np.pi/5) + 1j*np.sin(t*np.pi/5)/np.sqrt(5) * (a*numqi.gate.Y + b*numqi.gate.Z)
# physical_op = hfR(-2,-1,3)
logicalR = code.conj() @ hf_kron(*info['su2R']) @ code.T #hfR(-2,1)
print('\nlogical R(-2,1):', logicalR, sep='\n')

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T print('logical X:', logicalX, sep='\n') logicalZ5 = code.conj() @ hf_kron(*info['su2']) @ code.T print('\nlogical Z(2pi/5):', logicalZ5, sep='\n') # hfR = lambda a,b,t=1: numqi.gate.I*np.cos(t*np.pi/5) + 1j*np.sin(t*np.pi/5)/np.sqrt(5) * (a*numqi.gate.Y + b*numqi.gate.Z) # physical_op = hfR(-2,-1,3) logicalR = code.conj() @ hf_kron(*info['su2R']) @ code.T #hfR(-2,1) print('\nlogical R(-2,1):', logicalR, sep='\n')

logical X:
[[0.+0.j 1.+0.j]
 [1.+0.j 0.+0.j]]

logical Z(2pi/5):
[[0.80901699+0.58778525j 0.        +0.j        ]
 [0.        +0.j         0.80901699-0.58778525j]]

logical R(-2,1):
[[ 0.80901699+2.62865556e-01j -0.52573111-2.77555756e-17j]
 [ 0.52573111-2.77555756e-17j  0.80901699-2.62865556e-01j]]

2O lambda*=2¶

In [23]:

code,info = numqi.qec.q723.get_2O_X5(return_info=True)

print('code.shape:', code.shape)
print('Shor weight enumerator:', info['qweA'])
print('dual of Shor weight enumerator:', info['qweB'])

code,info = numqi.qec.q723.get_2O_X5(return_info=True) print('code.shape:', code.shape) print('Shor weight enumerator:', info['qweA']) print('dual of Shor weight enumerator:', info['qweB'])

code.shape: (2, 128)
Shor weight enumerator: [ 1  0  1  0 19  0 43  0]
dual of Shor weight enumerator: [  1   0   1  24  19 120  43  48]

In [24]:

logicalX = code.conj() @ hf_kron(*info['su2X']) @ code.T
print('logical X:', logicalX, sep='\n')

logicalYSY = code.conj() @ hf_kron(*info['su2YSY']) @ code.T
print('\nlogical Y(pi/4)SY(-pi/4):', logicalYSY, sep='\n')

logicalX = code.conj() @ hf_kron(*info['su2X']) @ code.T print('logical X:', logicalX, sep='\n') logicalYSY = code.conj() @ hf_kron(*info['su2YSY']) @ code.T print('\nlogical Y(pi/4)SY(-pi/4):', logicalYSY, sep='\n')

logical X:
[[0. 1.]
 [1. 0.]]

logical Y(pi/4)SY(-pi/4):
[[ 7.07106781e-01-0.5j  2.98181678e-18-0.5j]
 [-3.05576539e-18-0.5j  7.07106781e-01+0.5j]]

BD16, transversal T¶

$$ \begin{align*} 4\left|0_{L}\right\rangle =&e^{i\theta_{1}}\left|0000000\right\rangle +\sqrt{3}\left|0111100\right\rangle +\sqrt{3}\left|1001110\right\rangle \\ &+\sqrt{2}\left|1010101\right\rangle +\sqrt{2}\left|1011001\right\rangle +e^{i\theta_{2}}\left|1110010\right\rangle \\&+2i\left|0100011\right\rangle \\ \left|1_{L}\right\rangle =&X^{\otimes7}\left|0_{L}\right\rangle \end{align*} $$

In [25]:

theta = np_rng.uniform(0,2*np.pi,size=2)
sign = np_rng.integers(2, size=7)*2 - 1
code,info = numqi.qec.q723.get_BD16_veca1222233(*theta, sign=sign, return_info=True)

print('code.shape:', code.shape)
print('Shor weight enumerator:', np.around(info['qweA'],3))
print('dual of Shor weight enumerator:', np.around(info['qweB'],3))

theta = np_rng.uniform(0,2*np.pi,size=2) sign = np_rng.integers(2, size=7)*2 - 1 code,info = numqi.qec.q723.get_BD16_veca1222233(*theta, sign=sign, return_info=True) print('code.shape:', code.shape) print('Shor weight enumerator:', np.around(info['qweA'],3)) print('dual of Shor weight enumerator:', np.around(info['qweB'],3))

code.shape: (2, 128)
Shor weight enumerator: [ 1.     0.     2.625  0.325 12.72   7.139 38.137  2.054]
dual of Shor weight enumerator: [ 1.     0.     2.625 27.146 23.967 95.973 55.444 49.845]

In [26]:

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T
print('logical X:', logicalX, sep='\n')

logicalT = code.conj() @ hf_kron(*info['su2']) @ code.T
print('\nlogical T:', logicalT, sep='\n')

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T print('logical X:', logicalX, sep='\n') logicalT = code.conj() @ hf_kron(*info['su2']) @ code.T print('\nlogical T:', logicalT, sep='\n')

logical X:
[[0.+0.j 1.+0.j]
 [1.+0.j 0.+0.j]]

logical T:
[[0.92387953+0.38268343j 0.        +0.j        ]
 [0.        +0.j         0.92387953-0.38268343j]]

BD32, transversal sqrt(T)¶

In [27]:

sign = np_rng.integers(2, size=9)*2 - 1
code, info = numqi.qec.q723.get_BD32(np_rng.uniform(0,np.sqrt(1/8)), sign=sign, return_info=True)

print('code.shape:', code.shape)
print('Shor weight enumerator:', info['qweA'])
print('dual of Shor weight enumerator:', info['qweB'])

sign = np_rng.integers(2, size=9)*2 - 1 code, info = numqi.qec.q723.get_BD32(np_rng.uniform(0,np.sqrt(1/8)), sign=sign, return_info=True) print('code.shape:', code.shape) print('Shor weight enumerator:', info['qweA']) print('dual of Shor weight enumerator:', info['qweB'])

code.shape: (2, 128)
Shor weight enumerator: [ 1.          0.          2.08877259  0.         16.82245482  0.
 44.08877259  0.        ]
dual of Shor weight enumerator: [  1.           0.           2.08877259  27.26631777  16.82245482
 113.46736446  44.08877259  51.26631777]

In [28]:

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T
print('logical X:', logicalX, sep='\n')

logicalSqrtT = code.conj() @ hf_kron(*info['su2']) @ code.T
print('\nlogical Sqrt(T):', logicalSqrtT, sep='\n')

logicalX = code.conj() @ hf_kron(*[numqi.gate.X]*7) @ code.T print('logical X:', logicalX, sep='\n') logicalSqrtT = code.conj() @ hf_kron(*info['su2']) @ code.T print('\nlogical Sqrt(T):', logicalSqrtT, sep='\n')

logical X:
[[0.+0.j 1.+0.j]
 [1.+0.j 0.+0.j]]

logical Sqrt(T):
[[0.98078528+0.19509032j 0.        +0.j        ]
 [0.        +0.j         0.98078528-0.19509032j]]

More QECCs¶

Here we provides a list of available QECCs in numqi.qec:

In [29]:

tmp0 = [x for x in dir(numqi.qec.q723) if x.startswith('get_')]
for x in tmp0:
    print(f'numqi.qec.q723.{x}()')

tmp0 = [x for x in dir(numqi.qec.q723) if x.startswith('get_')] for x in tmp0: print(f'numqi.qec.q723.{x}()')

numqi.qec.q723.get_2I_lambda0()
numqi.qec.q723.get_2I_lambda075()
numqi.qec.q723.get_2O_X5()
numqi.qec.q723.get_723_cyclic_code_basis()
numqi.qec.q723.get_BD12_veca0122233()
numqi.qec.q723.get_BD12_veca1112222()
numqi.qec.q723.get_BD14()
numqi.qec.q723.get_BD16_5theta()
numqi.qec.q723.get_BD16_degenerate()
numqi.qec.q723.get_BD16_veca1222233()
numqi.qec.q723.get_BD18()
numqi.qec.q723.get_BD18_LP()
numqi.qec.q723.get_BD20()
numqi.qec.q723.get_BD22()
numqi.qec.q723.get_BD24()
numqi.qec.q723.get_BD26()
numqi.qec.q723.get_BD28()
numqi.qec.q723.get_BD30()
numqi.qec.q723.get_BD32()
numqi.qec.q723.get_BD34()
numqi.qec.q723.get_BD36()
numqi.qec.q723.get_cyclic_code()

In [30]:

tmp0 = [x for x in dir(numqi.qec.q823) if x.startswith('get_')]
for x in tmp0:
    print(f'numqi.qec.q823.{x}()')

tmp0 = [x for x in dir(numqi.qec.q823) if x.startswith('get_')] for x in tmp0: print(f'numqi.qec.q823.{x}()')

numqi.qec.q823.get_BD38_LP()
numqi.qec.q823.get_BD64()
numqi.qec.q823.get_BD72()
numqi.qec.q823.get_BD74()
numqi.qec.q823.get_BD76()
numqi.qec.q823.get_BD78()
numqi.qec.q823.get_BD80()
numqi.qec.q823.get_BD84()