Spin-½ Operators

This page details the specific operators available for spin-½ systems (qubits). These operators implement the general Operator interface described in the Operators concept guide.

Sites versus Support

All spin-½ operators possess at least one specific attribute: sites. It is crucial to distinguish this from the support of an operator.

• Sites: An ordered list of indices determining which qubits the operator targets when applied to a state. The order matters for operators that involve interactions (like CNOT, where the first site is target and the second is control) or for mapping matrix elements to qubits.
• Support: A set (unordered, unique) of indices representing the qubits involved in the operation.

For example, CNOT(1, 2) acting on sites 1 and 2 has sites = [1, 2] and support = {1, 2}.

var m = matrix([[0..4]], as_real)
CNOT(0, 1) * m // [[0, 1, 3, 2]]
CNOT(1, 0) * m // [[0, 3, 2, 1]]

Static Operators

Static operators are purely symbolic and require no numerical parameters, only the site indices they act upon. They represent common quantum gates and standard physics operators.

These are consistent in that they are lighter-weight objects defined entirely by their type and target sites.

Factory Function	Sites (Min)	Description
ID()	0	Identity operator (global)
X(site)	1	Pauli-X $\sigma^x$
Y(site)	1	Pauli-Y $\sigma^y$
Z(site)	1	Pauli-Z $\sigma^z$
SX(site)	1	Spin-X $S^x = \frac{1}{2}\sigma^x$
SY(site)	1	Spin-Y $S^y = \frac{1}{2}\sigma^y$
SZ(site)	1	Spin-Z $S^z = \frac{1}{2}\sigma^z$
Splus(site)	1	Raising operator $S^+ = S^x + iS^y$
Sminus(site)	1	Lowering operator $S^- = S^x - iS^y$
Proj0(site)	1	Projector onto $\| 0\rangle\langle 0 \|$
Proj1(site)	1	Projector onto $\| 1\rangle\langle 1 \|$
Hgate(site)	1	Hadamard gate
Sgate(site)	1	S gate ($P$ gate, $\sqrt{Z}$)
Tgate(site)	1	T gate ($\sqrt{S}$)
XX(site1, site2)	2	Ising interaction $X_i X_j$
YY(site1, site2)	2	Ising interaction $Y_i Y_j$
ZZ(site1, site2)	2	Ising interaction $Z_i Z_j$
XXPYY(site1, site2)	2	Exchange interaction $X_i X_j + Y_i Y_j$
CNOT(control, target)	2	Controlled-NOT gate
SWAP(site1, site2)	2	SWAP gate
ZN(sites)	N ($\ge 1$)	Sum of local Z terms: $\sum_{i} Z_i$
ZZNN(sites)	N ($\ge 2$)	Nearest-neighbour ZZ sum: $\sum_{i} Z_i Z_{i+1}$
ZZNNN(sites)	N ($\ge 3$)	Next-nearest-neighbour ZZ sum: $\sum_{i} Z_i Z_{i+2}$

Parametrized Operators

Parametrized operators are characterized by one or more continuous variables (typically real numbers), representing angles, phases, or coupling constants.

These operators are consistent as they all carry floating-point parameters defining their action (e.g. rotation angles).

Factory Function	Parameters	Sites	Description
Phase(phi)	phi (real)	0	Global phase $e^{-i\phi}$
RotX(phi, site)	phi (real)	1	Rotation around X: $e^{-i\frac{\phi}{2} X}$
RotY(phi, site)	phi (real)	1	Rotation around Y: $e^{-i\frac{\phi}{2} Y}$
RotZ(phi, site)	phi (real)	1	Rotation around Z: $e^{-i\frac{\phi}{2} Z}$
RotEuler(phi, theta, omega, site)	phi, theta, omega	1	$RZ(\omega)RX(\theta)RZ(\phi)$
UnitaryXYZ(Js, sites)	Js (vec3)	2	$e^{-i(J_x XX + J_y YY + J_z ZZ)}$
Floquet(Js, sites)	Js (vec3)	2	$e^{-i(J_x X + J_y Y + J_z Z) \otimes Z}$
DualUnitary(J, sites)	J (real)	2	$e^{i(\frac{\pi}{4} (XX + YY) + J ZZ)}$

Matrix Operators

Matrix operators are defined explicitly by their matrix elements. This category encompasses dense, sparse, and diagonal matrices.

These operators allow for arbitrary custom actions defined by the values in the matrix.

Factory Function	Parameters	Description
operator_dense(matrix, sites)	matrix, sites	Dense matrix operator.
operator_sparse(matrix, sites)	matrix, sites	Sparse matrix operator.
operator_diagonal(diag, sites)	diag, sites	Diagonal operator from vector.

Permutation Operators

Permutation operators rearrange the basis states of the system rather than mixing them via complex arithmetic.

These operators are efficient and consistent in that they represent index shuffles or bitwise operations on the state.

Factory Function	Parameters	Sites	Description
Flip(site)	None	1	Flips spin at site.
Flip(sites)	None	N	Flips spins at sites.
FlipAll()	None	Global	Flips all spins.
Reflect()	None	Global	Spatial reflection about center.
Translate(shift)	shift (int)	Global	Spatial translation.
Permute(p, sites)	p (vec), sites	N	Arbitrary permutation p on sites.

Pauli Strings

Pauli strings construct operators from sequences of Pauli matrices.

Consistency: Defined by string literals representing the tensor product structure.

Factory Function	Parameters	Description
pauli_string(word, sites)	word (string), sites	e.g. "XZ" applies X to first site, Z to second.

Generic Operators

Generic operators connect the high-level interface with user-defined low-level logic.

Consistency: Defined by a callback function f(matrix, indices) allowing arbitrary in-place modification.

Factory Function	Parameters	Description
operator_generic(func, sites)	func, sites	Applies func to state elements at sites.

Documentation Contributors

Jonathon Riddell

Eunji Yoo

Sebastien J Avakian

Vincent Michaud-Rioux