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equivalence_library.py
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equivalence_library.py
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# This code is part of Qiskit.
#
# (C) Copyright IBM 2017.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""Standard gates."""
from __future__ import annotations
from math import pi
from qiskit.circuit import (
EquivalenceLibrary,
Parameter,
QuantumCircuit,
QuantumRegister,
Gate,
Qubit,
Clbit,
)
from . import (
HGate,
CHGate,
PhaseGate,
CPhaseGate,
RGate,
RCCXGate,
RXGate,
CRXGate,
RXXGate,
RYGate,
CRYGate,
RZGate,
CRZGate,
RZZGate,
RZXGate,
SGate,
SdgGate,
CSGate,
CSdgGate,
SwapGate,
CSwapGate,
iSwapGate,
SXGate,
SXdgGate,
CSXGate,
DCXGate,
TGate,
TdgGate,
UGate,
CUGate,
U1Gate,
CU1Gate,
U2Gate,
U3Gate,
CU3Gate,
XGate,
CXGate,
CCXGate,
YGate,
CYGate,
RYYGate,
ECRGate,
ZGate,
CZGate,
IGate,
CCZGate,
XXPlusYYGate,
XXMinusYYGate,
)
_sel = StandardEquivalenceLibrary = EquivalenceLibrary()
def _cnot_rxx_decompose(plus_ry: bool = True, plus_rxx: bool = True):
"""Decomposition of CNOT gate.
NOTE: this differs to CNOT by a global phase.
The matrix returned is given by exp(1j * pi/4) * CNOT
Args:
plus_ry (bool): positive initial RY rotation
plus_rxx (bool): positive RXX rotation.
Returns:
QuantumCircuit: The decomposed circuit for CNOT gate (up to
global phase).
"""
# Convert boolean args to +/- 1 signs
if plus_ry:
sgn_ry = 1
else:
sgn_ry = -1
if plus_rxx:
sgn_rxx = 1
else:
sgn_rxx = -1
circuit = QuantumCircuit(2, global_phase=-sgn_ry * sgn_rxx * pi / 4)
circuit.append(RYGate(sgn_ry * pi / 2), [0])
circuit.append(RXXGate(sgn_rxx * pi / 2), [0, 1])
circuit.append(RXGate(-sgn_rxx * pi / 2), [0])
circuit.append(RXGate(-sgn_rxx * sgn_ry * pi / 2), [1])
circuit.append(RYGate(-sgn_ry * pi / 2), [0])
return circuit
# Import existing gate definitions
# HGate
#
# ┌───┐ ┌─────────┐
# q: ┤ H ├ ≡ q: ┤ U2(0,π) ├
# └───┘ └─────────┘
q = QuantumRegister(1, "q")
def_h = QuantumCircuit(q)
def_h.append(U2Gate(0, pi), [q[0]], [])
_sel.add_equivalence(HGate(), def_h)
# CHGate
#
# q_0: ──■── q_0: ─────────────────■─────────────────────
# ┌─┴─┐ ≡ ┌───┐┌───┐┌───┐┌─┴─┐┌─────┐┌───┐┌─────┐
# q_1: ┤ H ├ q_1: ┤ S ├┤ H ├┤ T ├┤ X ├┤ Tdg ├┤ H ├┤ Sdg ├
# └───┘ └───┘└───┘└───┘└───┘└─────┘└───┘└─────┘
q = QuantumRegister(2, "q")
def_ch = QuantumCircuit(q)
for inst, qargs, cargs in [
(SGate(), [q[1]], []),
(HGate(), [q[1]], []),
(TGate(), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(TdgGate(), [q[1]], []),
(HGate(), [q[1]], []),
(SdgGate(), [q[1]], []),
]:
def_ch.append(inst, qargs, cargs)
_sel.add_equivalence(CHGate(), def_ch)
# PhaseGate
#
# ┌──────┐ ┌───────┐
# q: ┤ P(ϴ) ├ ≡ q: ┤ U1(ϴ) ├
# └──────┘ └───────┘
q = QuantumRegister(1, "q")
theta = Parameter("theta")
phase_to_u1 = QuantumCircuit(q)
phase_to_u1.append(U1Gate(theta), [0])
_sel.add_equivalence(PhaseGate(theta), phase_to_u1)
q = QuantumRegister(1, "q")
theta = Parameter("theta")
phase_to_u = QuantumCircuit(q)
phase_to_u.u(0, 0, theta, 0)
_sel.add_equivalence(PhaseGate(theta), phase_to_u)
# CPhaseGate
# ┌────────┐
# q_0: ─■──── q_0: ┤ P(ϴ/2) ├──■───────────────■────────────
# │P(ϴ) ≡ └────────┘┌─┴─┐┌─────────┐┌─┴─┐┌────────┐
# q_1: ─■──── q_1: ──────────┤ X ├┤ P(-ϴ/2) ├┤ X ├┤ P(ϴ/2) ├
# └───┘└─────────┘└───┘└────────┘
q = QuantumRegister(2, "q")
theta = Parameter("theta")
def_cphase = QuantumCircuit(q)
def_cphase.p(theta / 2, 0)
def_cphase.cx(0, 1)
def_cphase.p(-theta / 2, 1)
def_cphase.cx(0, 1)
def_cphase.p(theta / 2, 1)
_sel.add_equivalence(CPhaseGate(theta), def_cphase)
# CPhaseGate
#
# q_0: ─■──── q_0: ─■────
# │P(ϴ) ≡ │U1(ϴ)
# q_1: ─■──── q_1: ─■────
q = QuantumRegister(2, "q")
theta = Parameter("theta")
cphase_to_cu1 = QuantumCircuit(q)
cphase_to_cu1.append(CU1Gate(theta), [0, 1])
_sel.add_equivalence(CPhaseGate(theta), cphase_to_cu1)
# RGate
#
# ┌────────┐ ┌───────────────────────┐
# q: ┤ R(ϴ,φ) ├ ≡ q: ┤ U3(ϴ,φ - π/2,π/2 - φ) ├
# └────────┘ └───────────────────────┘
q = QuantumRegister(1, "q")
theta = Parameter("theta")
phi = Parameter("phi")
def_r = QuantumCircuit(q)
def_r.append(U3Gate(theta, phi - pi / 2, -phi + pi / 2), [q[0]])
_sel.add_equivalence(RGate(theta, phi), def_r)
# IGate
q = QuantumRegister(1, "q")
def_id = QuantumCircuit(q)
def_id.append(UGate(0, 0, 0), [q[0]])
_sel.add_equivalence(IGate(), def_id)
q = QuantumRegister(1, "q")
def_id_rx = QuantumCircuit(q)
def_id_rx.append(RXGate(0), [q[0]])
_sel.add_equivalence(IGate(), def_id_rx)
q = QuantumRegister(1, "q")
def_id_ry = QuantumCircuit(q)
def_id_ry.append(RYGate(0), [q[0]])
_sel.add_equivalence(IGate(), def_id_ry)
q = QuantumRegister(1, "q")
def_id_rz = QuantumCircuit(q)
def_id_rz.append(RZGate(0), [q[0]])
_sel.add_equivalence(IGate(), def_id_rz)
# RCCXGate
#
# ┌───────┐
# q_0: ┤0 ├ q_0: ────────────────────────■────────────────────────
# │ │ │
# q_1: ┤1 Rccx ├ ≡ q_1: ────────────■───────────┼─────────■──────────────
# │ │ ┌───┐┌───┐┌─┴─┐┌─────┐┌─┴─┐┌───┐┌─┴─┐┌─────┐┌───┐
# q_2: ┤2 ├ q_2: ┤ H ├┤ T ├┤ X ├┤ Tdg ├┤ X ├┤ T ├┤ X ├┤ Tdg ├┤ H ├
# └───────┘ └───┘└───┘└───┘└─────┘└───┘└───┘└───┘└─────┘└───┘
q = QuantumRegister(3, "q")
def_rccx = QuantumCircuit(q)
for inst, qargs, cargs in [
(HGate(), [q[2]], []),
(TGate(), [q[2]], []),
(CXGate(), [q[1], q[2]], []),
(TdgGate(), [q[2]], []),
(CXGate(), [q[0], q[2]], []),
(TGate(), [q[2]], []),
(CXGate(), [q[1], q[2]], []),
(TdgGate(), [q[2]], []),
(HGate(), [q[2]], []),
]:
def_rccx.append(inst, qargs, cargs)
_sel.add_equivalence(RCCXGate(), def_rccx)
# RXGate
#
# ┌───────┐ ┌────────┐
# q: ┤ Rx(ϴ) ├ ≡ q: ┤ R(ϴ,0) ├
# └───────┘ └────────┘
q = QuantumRegister(1, "q")
theta = Parameter("theta")
def_rx = QuantumCircuit(q)
def_rx.append(RGate(theta, 0), [q[0]], [])
_sel.add_equivalence(RXGate(theta), def_rx)
# CRXGate
#
# q_0: ────■──── q_0: ─────────────■────────────────────■────────────────────
# ┌───┴───┐ ≡ ┌─────────┐┌─┴─┐┌──────────────┐┌─┴─┐┌────────────────┐
# q_1: ┤ Rx(ϴ) ├ q_1: ┤ U1(π/2) ├┤ X ├┤ U3(-ϴ/2,0,0) ├┤ X ├┤ U3(ϴ/2,-π/2,0) ├
# └───────┘ └─────────┘└───┘└──────────────┘└───┘└────────────────┘
q = QuantumRegister(2, "q")
theta = Parameter("theta")
def_crx = QuantumCircuit(q)
for inst, qargs, cargs in [
(U1Gate(pi / 2), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(U3Gate(-theta / 2, 0, 0), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(U3Gate(theta / 2, -pi / 2, 0), [q[1]], []),
]:
def_crx.append(inst, qargs, cargs)
_sel.add_equivalence(CRXGate(theta), def_crx)
# CRXGate
#
# q_0: ────■──── q_0: ───────■────────────────■────────────────────
# ┌───┴───┐ ≡ ┌───┐┌─┴─┐┌──────────┐┌─┴─┐┌─────────┐┌─────┐
# q_1: ┤ Rx(ϴ) ├ q_1: ┤ S ├┤ X ├┤ Ry(-ϴ/2) ├┤ X ├┤ Ry(ϴ/2) ├┤ Sdg ├
# └───────┘ └───┘└───┘└──────────┘└───┘└─────────┘└─────┘
q = QuantumRegister(2, "q")
theta = Parameter("theta")
crx_to_srycx = QuantumCircuit(q)
for inst, qargs, cargs in [
(SGate(), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(RYGate(-theta / 2), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(RYGate(theta / 2), [q[1]], []),
(SdgGate(), [q[1]], []),
]:
crx_to_srycx.append(inst, qargs, cargs)
_sel.add_equivalence(CRXGate(theta), crx_to_srycx)
# CRX in terms of one RXX
# ┌───┐ ┌────────────┐┌───┐
# q_0: ────■──── q_0: ───┤ H ├───┤0 ├┤ H ├
# ┌───┴───┐ ≡ ┌──┴───┴──┐│ Rxx(-ϴ/2) │└───┘
# q_1: ┤ Rx(ϴ) ├ q_1: ┤ Rx(ϴ/2) ├┤1 ├─────
# └───────┘ └─────────┘└────────────┘
theta = Parameter("theta")
crx_to_rxx = QuantumCircuit(2)
crx_to_rxx.h(0)
crx_to_rxx.rx(theta / 2, 1)
crx_to_rxx.rxx(-theta / 2, 0, 1)
crx_to_rxx.h(0)
_sel.add_equivalence(CRXGate(theta), crx_to_rxx)
# CRX to CRZ
#
# q_0: ────■──── q_0: ─────────■─────────
# ┌───┴───┐ ≡ ┌───┐┌───┴───┐┌───┐
# q_1: ┤ Rx(ϴ) ├ q_1: ┤ H ├┤ Rz(ϴ) ├┤ H ├
# └───────┘ └───┘└───────┘└───┘
theta = Parameter("theta")
crx_to_crz = QuantumCircuit(2)
crx_to_crz.h(1)
crx_to_crz.crz(theta, 0, 1)
crx_to_crz.h(1)
_sel.add_equivalence(CRXGate(theta), crx_to_crz)
# RXXGate
#
# ┌─────────┐ ┌───┐ ┌───┐
# q_0: ┤0 ├ q_0: ┤ H ├──■─────────────■──┤ H ├
# │ Rxx(ϴ) │ ≡ ├───┤┌─┴─┐┌───────┐┌─┴─┐├───┤
# q_1: ┤1 ├ q_1: ┤ H ├┤ X ├┤ Rz(ϴ) ├┤ X ├┤ H ├
# └─────────┘ └───┘└───┘└───────┘└───┘└───┘
q = QuantumRegister(2, "q")
theta = Parameter("theta")
def_rxx = QuantumCircuit(q)
for inst, qargs, cargs in [
(HGate(), [q[0]], []),
(HGate(), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(RZGate(theta), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(HGate(), [q[1]], []),
(HGate(), [q[0]], []),
]:
def_rxx.append(inst, qargs, cargs)
_sel.add_equivalence(RXXGate(theta), def_rxx)
# RXX to RZX
# ┌─────────┐ ┌───┐┌─────────┐┌───┐
# q_0: ┤0 ├ q_0: ┤ H ├┤0 ├┤ H ├
# │ Rxx(ϴ) │ ≡ └───┘│ Rzx(ϴ) │└───┘
# q_1: ┤1 ├ q_1: ─────┤1 ├─────
# └─────────┘ └─────────┘
theta = Parameter("theta")
rxx_to_rzx = QuantumCircuit(2)
rxx_to_rzx.h(0)
rxx_to_rzx.rzx(theta, 0, 1)
rxx_to_rzx.h(0)
_sel.add_equivalence(RXXGate(theta), rxx_to_rzx)
# RXX to RZZ
q = QuantumRegister(2, "q")
theta = Parameter("theta")
rxx_to_rzz = QuantumCircuit(q)
for inst, qargs, cargs in [
(HGate(), [q[0]], []),
(HGate(), [q[1]], []),
(RZZGate(theta), [q[0], q[1]], []),
(HGate(), [q[0]], []),
(HGate(), [q[1]], []),
]:
rxx_to_rzz.append(inst, qargs, cargs)
_sel.add_equivalence(RXXGate(theta), rxx_to_rzz)
# RZXGate
#
# ┌─────────┐
# q_0: ┤0 ├ q_0: ───────■─────────────■───────
# │ Rzx(ϴ) │ ≡ ┌───┐┌─┴─┐┌───────┐┌─┴─┐┌───┐
# q_1: ┤1 ├ q_1: ┤ H ├┤ X ├┤ Rz(ϴ) ├┤ X ├┤ H ├
# └─────────┘ └───┘└───┘└───────┘└───┘└───┘
q = QuantumRegister(2, "q")
theta = Parameter("theta")
def_rzx = QuantumCircuit(q)
for inst, qargs, cargs in [
(HGate(), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(RZGate(theta), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(HGate(), [q[1]], []),
]:
def_rzx.append(inst, qargs, cargs)
_sel.add_equivalence(RZXGate(theta), def_rzx)
# RYGate
#
# ┌───────┐ ┌──────────┐
# q: ┤ Ry(ϴ) ├ ≡ q: ┤ R(ϴ,π/2) ├
# └───────┘ └──────────┘
q = QuantumRegister(1, "q")
theta = Parameter("theta")
def_ry = QuantumCircuit(q)
def_ry.append(RGate(theta, pi / 2), [q[0]], [])
_sel.add_equivalence(RYGate(theta), def_ry)
q = QuantumRegister(1, "q")
ry_to_rx = QuantumCircuit(q)
ry_to_rx.sdg(0)
ry_to_rx.rx(theta, 0)
ry_to_rx.s(0)
_sel.add_equivalence(RYGate(theta), ry_to_rx)
# CRYGate
#
# q_0: ────■──── q_0: ─────────────■────────────────■──
# ┌───┴───┐ ≡ ┌─────────┐┌─┴─┐┌──────────┐┌─┴─┐
# q_1: ┤ Ry(ϴ) ├ q_1: ┤ Ry(ϴ/2) ├┤ X ├┤ Ry(-ϴ/2) ├┤ X ├
# └───────┘ └─────────┘└───┘└──────────┘└───┘
q = QuantumRegister(2, "q")
theta = Parameter("theta")
def_cry = QuantumCircuit(q)
for inst, qargs, cargs in [
(RYGate(theta / 2), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(RYGate(-theta / 2), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
]:
def_cry.append(inst, qargs, cargs)
_sel.add_equivalence(CRYGate(theta), def_cry)
# CRY to CRZ
#
# q_0: ────■──── q_0: ───────────────■────────────────
# ┌───┴───┐ ≡ ┌─────────┐┌───┴───┐┌──────────┐
# q_1: ┤ Ry(ϴ) ├ q_1: ┤ Rx(π/2) ├┤ Rz(ϴ) ├┤ Rx(-π/2) ├
# └───────┘ └─────────┘└───────┘└──────────┘
theta = Parameter("theta")
cry_to_crz = QuantumCircuit(2)
cry_to_crz.rx(pi / 2, 1)
cry_to_crz.crz(theta, 0, 1)
cry_to_crz.rx(-pi / 2, 1)
_sel.add_equivalence(CRYGate(theta), cry_to_crz)
# CRY to CRZ
#
# q_0: ────■──── q_0: ────────────────────■─────────────────────
# ┌───┴───┐ ≡ ┌───┐┌─────────┐┌───┴───┐┌──────────┐┌───┐
# q_1: ┤ Ry(ϴ) ├ q_1: ┤ H ├┤ Rz(π/2) ├┤ Rx(ϴ) ├┤ Rz(-π/2) ├┤ H ├
# └───────┘ └───┘└─────────┘└───────┘└──────────┘└───┘
theta = Parameter("theta")
cry_to_crx = QuantumCircuit(2)
cry_to_crx.h(1)
cry_to_crx.rz(pi / 2, 1)
cry_to_crx.crx(theta, 0, 1)
cry_to_crx.rz(-pi / 2, 1)
cry_to_crx.h(1)
_sel.add_equivalence(CRYGate(theta), cry_to_crx)
# CRY to RZZ
#
# q_0: ────■──── q_0: ────────────────────────■───────────────────
# ┌───┴───┐ ≡ ┌─────┐┌─────────┐┌───┐ │ZZ(-ϴ/2) ┌───┐┌───┐
# q_1: ┤ Ry(ϴ) ├ q_1: ┤ Sdg ├┤ Rx(ϴ/2) ├┤ H ├─■─────────┤ H ├┤ S ├
# └───────┘ └─────┘└─────────┘└───┘ └───┘└───┘
cry_to_rzz = QuantumCircuit(2)
cry_to_rzz.sdg(1)
cry_to_rzz.rx(theta / 2, 1)
cry_to_rzz.h(1)
cry_to_rzz.rzz(-theta / 2, 0, 1)
cry_to_rzz.h(1)
cry_to_rzz.s(1)
_sel.add_equivalence(CRYGate(theta), cry_to_rzz)
# RYYGate
#
# ┌─────────┐ ┌─────────┐ ┌──────────┐
# q_0: ┤0 ├ q_0: ┤ Rx(π/2) ├──■─────────────■──┤ Rx(-π/2) ├
# │ Ryy(ϴ) │ ≡ ├─────────┤┌─┴─┐┌───────┐┌─┴─┐├──────────┤
# q_1: ┤1 ├ q_1: ┤ Rx(π/2) ├┤ X ├┤ Rz(ϴ) ├┤ X ├┤ Rx(-π/2) ├
# └─────────┘ └─────────┘└───┘└───────┘└───┘└──────────┘
q = QuantumRegister(2, "q")
theta = Parameter("theta")
def_ryy = QuantumCircuit(q)
for inst, qargs, cargs in [
(RXGate(pi / 2), [q[0]], []),
(RXGate(pi / 2), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(RZGate(theta), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(RXGate(-pi / 2), [q[0]], []),
(RXGate(-pi / 2), [q[1]], []),
]:
def_ryy.append(inst, qargs, cargs)
_sel.add_equivalence(RYYGate(theta), def_ryy)
# RYY to RZZ
q = QuantumRegister(2, "q")
theta = Parameter("theta")
ryy_to_rzz = QuantumCircuit(q)
for inst, qargs, cargs in [
(RXGate(pi / 2), [q[0]], []),
(RXGate(pi / 2), [q[1]], []),
(RZZGate(theta), [q[0], q[1]], []),
(RXGate(-pi / 2), [q[0]], []),
(RXGate(-pi / 2), [q[1]], []),
]:
ryy_to_rzz.append(inst, qargs, cargs)
_sel.add_equivalence(RYYGate(theta), ryy_to_rzz)
# RYY to RXX
q = QuantumRegister(2, "q")
theta = Parameter("theta")
ryy_to_rxx = QuantumCircuit(q)
for inst, qargs, cargs in [
(SdgGate(), [q[0]], []),
(SdgGate(), [q[1]], []),
(RXXGate(theta), [q[0], q[1]], []),
(SGate(), [q[0]], []),
(SGate(), [q[1]], []),
]:
ryy_to_rxx.append(inst, qargs, cargs)
_sel.add_equivalence(RYYGate(theta), ryy_to_rxx)
# RZGate
# global phase: -ϴ/2
# ┌───────┐ ┌───────┐
# q: ┤ Rz(ϴ) ├ ≡ q: ┤ U1(ϴ) ├
# └───────┘ └───────┘
q = QuantumRegister(1, "q")
theta = Parameter("theta")
def_rz = QuantumCircuit(q, global_phase=-theta / 2)
def_rz.append(U1Gate(theta), [q[0]], [])
_sel.add_equivalence(RZGate(theta), def_rz)
# RZGate
#
# ┌───────┐ ┌────┐┌────────┐┌──────┐
# q: ┤ Rz(ϴ) ├ ≡ q: ┤ √X ├┤ Ry(-ϴ) ├┤ √Xdg ├
# └───────┘ └────┘└────────┘└──────┘
q = QuantumRegister(1, "q")
rz_to_sxry = QuantumCircuit(q)
rz_to_sxry.sx(0)
rz_to_sxry.ry(-theta, 0)
rz_to_sxry.sxdg(0)
_sel.add_equivalence(RZGate(theta), rz_to_sxry)
q = QuantumRegister(1, "q")
rz_to_rx = QuantumCircuit(q)
rz_to_rx.h(0)
rz_to_rx.rx(theta, 0)
rz_to_rx.h(0)
_sel.add_equivalence(RZGate(theta), rz_to_rx)
# CRZGate
#
# q_0: ────■──── q_0: ─────────────■────────────────■──
# ┌───┴───┐ ≡ ┌─────────┐┌─┴─┐┌──────────┐┌─┴─┐
# q_1: ┤ Rz(ϴ) ├ q_1: ┤ Rz(ϴ/2) ├┤ X ├┤ Rz(-ϴ/2) ├┤ X ├
# └───────┘ └─────────┘└───┘└──────────┘└───┘
q = QuantumRegister(2, "q")
theta = Parameter("theta")
def_crz = QuantumCircuit(q)
for inst, qargs, cargs in [
(RZGate(theta / 2), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(RZGate(-theta / 2), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
]:
def_crz.append(inst, qargs, cargs)
_sel.add_equivalence(CRZGate(theta), def_crz)
# CRZ to CRY
#
# q_0: ────■──── q_0: ────────────────■───────────────
# ┌───┴───┐ ≡ ┌──────────┐┌───┴───┐┌─────────┐
# q_1: ┤ Rz(ϴ) ├ q_1: ┤ Rx(-π/2) ├┤ Ry(ϴ) ├┤ Rx(π/2) ├
# └───────┘ └──────────┘└───────┘└─────────┘
theta = Parameter("theta")
crz_to_cry = QuantumCircuit(2)
crz_to_cry.rx(-pi / 2, 1)
crz_to_cry.cry(theta, 0, 1)
crz_to_cry.rx(pi / 2, 1)
_sel.add_equivalence(CRZGate(theta), crz_to_cry)
# CRZ to CRX
#
# q_0: ────■──── q_0: ─────────■─────────
# ┌───┴───┐ ≡ ┌───┐┌───┴───┐┌───┐
# q_1: ┤ Rz(ϴ) ├ q_1: ┤ H ├┤ Rx(ϴ) ├┤ H ├
# └───────┘ └───┘└───────┘└───┘
theta = Parameter("theta")
crz_to_crx = QuantumCircuit(2)
crz_to_crx.h(1)
crz_to_crx.crx(theta, 0, 1)
crz_to_crx.h(1)
_sel.add_equivalence(CRZGate(theta), crz_to_crx)
# CRZ to RZZ
#
# q_0: ────■──── q_0: ────────────■────────
# ┌───┴───┐ ≡ ┌─────────┐ │ZZ(-ϴ/2)
# q_1: ┤ Rz(ϴ) ├ q_1: ┤ Rz(ϴ/2) ├─■────────
# └───────┘ └─────────┘
theta = Parameter("theta")
crz_to_rzz = QuantumCircuit(2)
crz_to_rzz.rz(theta / 2, 1)
crz_to_rzz.rzz(-theta / 2, 0, 1)
_sel.add_equivalence(CRZGate(theta), crz_to_rzz)
# RZZGate
#
# q_0: ─■───── q_0: ──■─────────────■──
# │ZZ(ϴ) ≡ ┌─┴─┐┌───────┐┌─┴─┐
# q_1: ─■───── q_1: ┤ X ├┤ Rz(ϴ) ├┤ X ├
# └───┘└───────┘└───┘
q = QuantumRegister(2, "q")
theta = Parameter("theta")
def_rzz = QuantumCircuit(q)
for inst, qargs, cargs in [
(CXGate(), [q[0], q[1]], []),
(RZGate(theta), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
]:
def_rzz.append(inst, qargs, cargs)
_sel.add_equivalence(RZZGate(theta), def_rzz)
# RZZ to RXX
q = QuantumRegister(2, "q")
theta = Parameter("theta")
rzz_to_rxx = QuantumCircuit(q)
for inst, qargs, cargs in [
(HGate(), [q[0]], []),
(HGate(), [q[1]], []),
(RXXGate(theta), [q[0], q[1]], []),
(HGate(), [q[0]], []),
(HGate(), [q[1]], []),
]:
rzz_to_rxx.append(inst, qargs, cargs)
_sel.add_equivalence(RZZGate(theta), rzz_to_rxx)
# RZZ to RZX
# ┌─────────┐
# q_0: ─■───── q_0: ─────┤0 ├─────
# │ZZ(ϴ) ≡ ┌───┐│ Rzx(ϴ) │┌───┐
# q_1: ─■───── q_1: ┤ H ├┤1 ├┤ H ├
# └───┘└─────────┘└───┘
theta = Parameter("theta")
rzz_to_rzx = QuantumCircuit(2)
rzz_to_rzx.h(1)
rzz_to_rzx.rzx(theta, 0, 1)
rzz_to_rzx.h(1)
_sel.add_equivalence(RZZGate(theta), rzz_to_rzx)
# RZZ to RYY
q = QuantumRegister(2, "q")
theta = Parameter("theta")
rzz_to_ryy = QuantumCircuit(q)
for inst, qargs, cargs in [
(RXGate(-pi / 2), [q[0]], []),
(RXGate(-pi / 2), [q[1]], []),
(RYYGate(theta), [q[0], q[1]], []),
(RXGate(pi / 2), [q[0]], []),
(RXGate(pi / 2), [q[1]], []),
]:
rzz_to_ryy.append(inst, qargs, cargs)
_sel.add_equivalence(RZZGate(theta), rzz_to_ryy)
# RZXGate
#
# ┌─────────┐
# q_0: ┤0 ├ q_0: ───────■─────────────■───────
# │ Rzx(ϴ) │ ≡ ┌───┐┌─┴─┐┌───────┐┌─┴─┐┌───┐
# q_1: ┤1 ├ q_1: ┤ H ├┤ X ├┤ Rz(ϴ) ├┤ X ├┤ H ├
# └─────────┘ └───┘└───┘└───────┘└───┘└───┘
q = QuantumRegister(2, "q")
theta = Parameter("theta")
def_rzx = QuantumCircuit(q)
for inst, qargs, cargs in [
(HGate(), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(RZGate(theta), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(HGate(), [q[1]], []),
]:
def_rzx.append(inst, qargs, cargs)
_sel.add_equivalence(RZXGate(theta), def_rzx)
# ECRGate
#
# ┌──────┐ ┌───────────┐┌───┐┌────────────┐
# q_0: ┤0 ├ q_0: ┤0 ├┤ X ├┤0 ├
# │ Ecr │ ≡ │ Rzx(π/4) │└───┘│ Rzx(-π/4) │
# q_1: ┤1 ├ q_1: ┤1 ├─────┤1 ├
# └──────┘ └───────────┘ └────────────┘
q = QuantumRegister(2, "q")
def_ecr = QuantumCircuit(q)
for inst, qargs, cargs in [
(RZXGate(pi / 4), [q[0], q[1]], []),
(XGate(), [q[0]], []),
(RZXGate(-pi / 4), [q[0], q[1]], []),
]:
def_ecr.append(inst, qargs, cargs)
_sel.add_equivalence(ECRGate(), def_ecr)
# ECRGate decomposed to Clifford gates (up to a global phase)
#
# global phase: 7π/4
# ┌──────┐ ┌───┐ ┌───┐
# q_0: ┤0 ├ q_0: ┤ S ├───■──┤ X ├
# │ Ecr │ ≡ ├───┴┐┌─┴─┐└───┘
# q_1: ┤1 ├ q_1: ┤ √X ├┤ X ├─────
# └──────┘ └────┘└───┘
q = QuantumRegister(2, "q")
def_ecr_cliff = QuantumCircuit(q, global_phase=-pi / 4)
for inst, qargs, cargs in [
(SGate(), [q[0]], []),
(SXGate(), [q[1]], []),
(CXGate(), [q[0], q[1]], []),
(XGate(), [q[0]], []),
]:
def_ecr_cliff.append(inst, qargs, cargs)
_sel.add_equivalence(ECRGate(), def_ecr_cliff)
# CXGate decomposed using an ECRGate and Clifford 1-qubit gates
# global phase: π/4
# q_0: ──■── ┌─────┐ ┌──────┐┌───┐
# ┌─┴─┐ ≡ q_0: ┤ Sdg ├─┤0 ├┤ X ├
# q_1: ┤ X ├ ├─────┴┐│ Ecr │└───┘
# └───┘ q_1: ┤ √Xdg ├┤1 ├─────
# └──────┘└──────┘
q = QuantumRegister(2, "q")
def_ecr_to_cx_cliff = QuantumCircuit(q, global_phase=pi / 4)
for inst, qargs, cargs in [
(SdgGate(), [q[0]], []),
(SXdgGate(), [q[1]], []),
(ECRGate(), [q[0], q[1]], []),
(XGate(), [q[0]], []),
]:
def_ecr_to_cx_cliff.append(inst, qargs, cargs)
_sel.add_equivalence(CXGate(), def_ecr_to_cx_cliff)
# SGate
#
# ┌───┐ ┌─────────┐
# q: ┤ S ├ ≡ q: ┤ U1(π/2) ├
# └───┘ └─────────┘
q = QuantumRegister(1, "q")
def_s = QuantumCircuit(q)
def_s.append(U1Gate(pi / 2), [q[0]], [])
_sel.add_equivalence(SGate(), def_s)
# SdgGate
#
# ┌─────┐ ┌──────────┐
# q: ┤ Sdg ├ ≡ q: ┤ U1(-π/2) ├
# └─────┘ └──────────┘
q = QuantumRegister(1, "q")
def_sdg = QuantumCircuit(q)
def_sdg.append(U1Gate(-pi / 2), [q[0]], [])
_sel.add_equivalence(SdgGate(), def_sdg)
# CSGate
#
# q_0: ──■── q_0: ───────■────────
# ┌─┴─┐ ┌───┐┌─┴──┐┌───┐
# q_1: ┤ S ├ = q_1: ┤ H ├┤ Sx ├┤ H ├
# └───┘ └───┘└────┘└───┘
q = QuantumRegister(2, "q")
def_cs = QuantumCircuit(q)
def_cs.append(HGate(), [q[1]], [])
def_cs.append(CSXGate(), [q[0], q[1]], [])
def_cs.append(HGate(), [q[1]], [])
_sel.add_equivalence(CSGate(), def_cs)
# CSdgGate
#
# q_0: ───■─── q_0: ───────■────■────────
# ┌──┴──┐ ┌───┐┌─┴─┐┌─┴──┐┌───┐
# q_1: ┤ Sdg ├ = q_1: ┤ H ├┤ X ├┤ Sx ├┤ H ├
# └─────┘ └───┘└───┘└────┘└───┘
q = QuantumRegister(2, "q")
def_csdg = QuantumCircuit(q)
def_csdg.append(HGate(), [q[1]], [])
def_csdg.append(CXGate(), [q[0], q[1]], [])
def_csdg.append(CSXGate(), [q[0], q[1]], [])
def_csdg.append(HGate(), [q[1]], [])
_sel.add_equivalence(CSdgGate(), def_csdg)
# SdgGate
#
# ┌─────┐ ┌───┐┌───┐
# q: ┤ Sdg ├ ≡ q: ┤ S ├┤ Z ├
# └─────┘ └───┘└───┘
q = QuantumRegister(1, "q")
def_sdg = QuantumCircuit(q)
for inst, qargs, cargs in [
(SGate(), [q[0]], []),
(ZGate(), [q[0]], []),
]:
def_sdg.append(inst, qargs, cargs)
_sel.add_equivalence(SdgGate(), def_sdg)
# SdgGate
#
# ┌─────┐ ┌───┐┌───┐
# q: ┤ Sdg ├ ≡ q: ┤ Z ├┤ S ├
# └─────┘ └───┘└───┘
q = QuantumRegister(1, "q")
def_sdg = QuantumCircuit(q)
for inst, qargs, cargs in [
(ZGate(), [q[0]], []),
(SGate(), [q[0]], []),
]:
def_sdg.append(inst, qargs, cargs)
_sel.add_equivalence(SdgGate(), def_sdg)
# SdgGate
#
# ┌─────┐ ┌───┐┌───┐┌───┐
# q: ┤ Sdg ├ ≡ q: ┤ S ├┤ S ├┤ S ├
# └─────┘ └───┘└───┘└───┘
q = QuantumRegister(1, "q")
def_sdg = QuantumCircuit(q)
for inst, qargs, cargs in [
(SGate(), [q[0]], []),
(SGate(), [q[0]], []),
(SGate(), [q[0]], []),
]:
def_sdg.append(inst, qargs, cargs)
_sel.add_equivalence(SdgGate(), def_sdg)
# SwapGate
# ┌───┐
# q_0: ─X─ q_0: ──■──┤ X ├──■──
# │ ≡ ┌─┴─┐└─┬─┘┌─┴─┐
# q_1: ─X─ q_1: ┤ X ├──■──┤ X ├
# └───┘ └───┘
q = QuantumRegister(2, "q")
def_swap = QuantumCircuit(q)
for inst, qargs, cargs in [
(CXGate(), [q[0], q[1]], []),
(CXGate(), [q[1], q[0]], []),
(CXGate(), [q[0], q[1]], []),
]:
def_swap.append(inst, qargs, cargs)
_sel.add_equivalence(SwapGate(), def_swap)
# SwapGate
#
# q_0: ─X─
# │ ≡
# q_1: ─X─
#
# ┌──────────┐┌──────┐ ┌────┐ ┌──────┐┌──────────┐┌──────┐
# q_0: ┤ Rz(-π/2) ├┤0 ├───┤ √X ├───┤1 ├┤ Rz(-π/2) ├┤0 ├
# └──┬────┬──┘│ Ecr │┌──┴────┴──┐│ Ecr │└──┬────┬──┘│ Ecr │
# q_1: ───┤ √X ├───┤1 ├┤ Rz(-π/2) ├┤0 ├───┤ √X ├───┤1 ├
# └────┘ └──────┘└──────────┘└──────┘ └────┘ └──────┘
#
q = QuantumRegister(2, "q")
def_swap_ecr = QuantumCircuit(q)
def_swap_ecr.rz(-pi / 2, 0)
def_swap_ecr.sx(1)
def_swap_ecr.ecr(0, 1)
def_swap_ecr.rz(-pi / 2, 1)
def_swap_ecr.sx(0)
def_swap_ecr.ecr(1, 0)
def_swap_ecr.rz(-pi / 2, 0)
def_swap_ecr.sx(1)
def_swap_ecr.ecr(0, 1)
_sel.add_equivalence(SwapGate(), def_swap_ecr)
# SwapGate
#
# q_0: ─X─
# │ ≡
# q_1: ─X─
#
# global phase: 3π/2
# ┌────┐ ┌────┐ ┌────┐
# q_0: ┤ √X ├─■─┤ √X ├─■─┤ √X ├─■─
# ├────┤ │ ├────┤ │ ├────┤ │
# q_1: ┤ √X ├─■─┤ √X ├─■─┤ √X ├─■─
# └────┘ └────┘ └────┘
q = QuantumRegister(2, "q")
def_swap_cz = QuantumCircuit(q, global_phase=-pi / 2)
def_swap_cz.sx(0)
def_swap_cz.sx(1)
def_swap_cz.cz(0, 1)
def_swap_cz.sx(0)
def_swap_cz.sx(1)
def_swap_cz.cz(0, 1)
def_swap_cz.sx(0)
def_swap_cz.sx(1)
def_swap_cz.cz(0, 1)
_sel.add_equivalence(SwapGate(), def_swap_cz)
# iSwapGate
#
# ┌────────┐ ┌───┐┌───┐ ┌───┐
# q_0: ┤0 ├ q_0: ┤ S ├┤ H ├──■──┤ X ├─────
# │ Iswap │ ≡ ├───┤└───┘┌─┴─┐└─┬─┘┌───┐
# q_1: ┤1 ├ q_1: ┤ S ├─────┤ X ├──■──┤ H ├
# └────────┘ └───┘ └───┘ └───┘
q = QuantumRegister(2, "q")
def_iswap = QuantumCircuit(q)
for inst, qargs, cargs in [
(SGate(), [q[0]], []),
(SGate(), [q[1]], []),
(HGate(), [q[0]], []),
(CXGate(), [q[0], q[1]], []),
(CXGate(), [q[1], q[0]], []),
(HGate(), [q[1]], []),
]:
def_iswap.append(inst, qargs, cargs)
_sel.add_equivalence(iSwapGate(), def_iswap)
# SXGate
# global phase: π/4
# ┌────┐ ┌─────┐┌───┐┌─────┐
# q: ┤ √X ├ ≡ q: ┤ Sdg ├┤ H ├┤ Sdg ├
# └────┘ └─────┘└───┘└─────┘
q = QuantumRegister(1, "q")
def_sx = QuantumCircuit(q, global_phase=pi / 4)
for inst, qargs, cargs in [(SdgGate(), [q[0]], []), (HGate(), [q[0]], []), (SdgGate(), [q[0]], [])]:
def_sx.append(inst, qargs, cargs)
_sel.add_equivalence(SXGate(), def_sx)
# HGate decomposed into SXGate and SGate
# global phase: -π/4
# ┌───┐ ┌───┐┌────┐┌───┐
# q: ┤ H ├ ≡ q: ┤ S ├┤ √X ├┤ S ├
# └───┘ └───┘└────┘└───┘
q = QuantumRegister(1, "q")
def_h_to_sx = QuantumCircuit(q, global_phase=-pi / 4)
for inst, qargs, cargs in [(SGate(), [q[0]], []), (SXGate(), [q[0]], []), (SGate(), [q[0]], [])]:
def_h_to_sx.append(inst, qargs, cargs)
_sel.add_equivalence(HGate(), def_h_to_sx)
# SXGate
# global phase: π/4
# ┌────┐ ┌─────────┐
# q: ┤ √X ├ ≡ q: ┤ Rx(π/2) ├
# └────┘ └─────────┘
q = QuantumRegister(1, "q")
sx_to_rx = QuantumCircuit(q, global_phase=pi / 4)
sx_to_rx.rx(pi / 2, 0)
_sel.add_equivalence(SXGate(), sx_to_rx)
# SXdgGate
# global phase: 7π/4
# ┌──────┐ ┌───┐┌───┐┌───┐
# q: ┤ √Xdg ├ ≡ q: ┤ S ├┤ H ├┤ S ├
# └──────┘ └───┘└───┘└───┘
q = QuantumRegister(1, "q")
def_sxdg = QuantumCircuit(q, global_phase=-pi / 4)
for inst, qargs, cargs in [(SGate(), [q[0]], []), (HGate(), [q[0]], []), (SGate(), [q[0]], [])]:
def_sxdg.append(inst, qargs, cargs)
_sel.add_equivalence(SXdgGate(), def_sxdg)
# HGate decomposed into SXdgGate and SdgGate
# global phase: π/4
# ┌───┐ ┌─────┐┌──────┐┌─────┐
# q: ┤ H ├ ≡ q: ┤ Sdg ├┤ √Xdg ├┤ Sdg ├
# └───┘ └─────┘└──────┘└─────┘
q = QuantumRegister(1, "q")
def_h_to_sxdg = QuantumCircuit(q, global_phase=pi / 4)
for inst, qargs, cargs in [
(SdgGate(), [q[0]], []),
(SXdgGate(), [q[0]], []),
(SdgGate(), [q[0]], []),
]:
def_h_to_sxdg.append(inst, qargs, cargs)
_sel.add_equivalence(HGate(), def_h_to_sxdg)
# SXdgGate
# global phase: 7π/4
# ┌──────┐ ┌──────────┐
# q: ┤ √Xdg ├ ≡ q: ┤ Rx(-π/2) ├
# └──────┘ └──────────┘
q = QuantumRegister(1, "q")
sxdg_to_rx = QuantumCircuit(q, global_phase=-pi / 4)
sxdg_to_rx.rx(-pi / 2, 0)
_sel.add_equivalence(SXdgGate(), sxdg_to_rx)
# CSXGate
#
# q_0: ──■─── q_0: ──────■─────────────
# ┌─┴──┐ ≡ ┌───┐ │U1(π/2) ┌───┐
# q_1: ┤ Sx ├ q_1: ┤ H ├─■────────┤ H ├
# └────┘ └───┘ └───┘
q = QuantumRegister(2, "q")
def_csx = QuantumCircuit(q)
for inst, qargs, cargs in [
(HGate(), [q[1]], []),