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description= What each quantum gate actually does to a qubit state, with circuit diagrams and matrix forms for Hadamard, Pauli-X/Y/Z, CNOT, Toffoli, and the…;
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the (243), gate (142), gates (120), and (106), quantum (90), cnot (86), qubit (77), #circuit (48), are (46), rangle (46), with (44), this (41), for (40), two (38), from (36), quantumcircuit (35), clifford (34), qiskit (33), qubits (30), you (30), import (30), error (28), not (28), that (28), control (28), phase (26), print (26), target (26), text (24), hadamard (21), controlled (21), iswap (21), can (20), single (20), state (19), array (18), theta (18), hardware (17), native (17), rotation (17), toffoli (17), pauli (17), all (16), measurement (16), cdot (16), identity (15), fidelity (15), when (15), states (15), computing (14), operation (14), classical (14), otimes (14), into (14), matrix (14), superconducting (14), ibm (13), read (13), any (13), courses (12), how (12), set (12), params (12), dagger (12), pulse (12), cos (12), sin (12), use (11), time (11), numpy (11), applies (11), both (11), each (11), guide (10), one (10), where (10), variational (10), algorithms (10), noise (10), 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Text of the page (random words):
z np allclose s z s conj t z true for cnot the pauli propagation rules involve both qubits cnot x i cnot x x text cnot cdot x otimes i cdot text cnot dagger x otimes x cnot x i cnot x x cnot i x cnot i x text cnot cdot i otimes x cdot text cnot dagger i otimes x cnot i x cnot i x cnot z i cnot z i text cnot cdot z otimes i cdot text cnot dagger z otimes i cnot z i cnot z i cnot i z cnot z z text cnot cdot i otimes z cdot text cnot dagger z otimes z cnot i z cnot z z the first rule is particularly important an x error on the control qubit spreads to the target qubit through a cnot understanding how errors propagate through clifford gates is central to quantum error correction why clifford gates are special the gottesman knill theorem the gottesman knill theorem states that any quantum circuit composed entirely of clifford gates starting from computational basis states with measurements only in the computational basis can be efficiently simulated on a classical computer efficiently means in polynomial time and space regardless of the number of qubits this is a profound result it means that entanglement alone is not sufficient for quantum speedup a bell state circuit h cnot is a clifford circuit and is classically simulable quantum advantage requires non clifford gates and the t gate is the simplest non clifford gate you can verify that t is not clifford by checking its conjugation of x t x t 1 2 x y t x t dagger frac 1 sqrt 2 x y t x t 2 1 x y this is not a pauli operator the t gate maps paulis outside the pauli group so t is not a member of the clifford group ancilla qubits and gate decomposition an ancilla qubit is a helper qubit that starts in a known state participates in a computation and is either restored to its original state or measured and discarded ancilla qubits enable the construction of complex multi qubit gates from simpler operations the toffoli decomposition the toffoli gate is a three qubit gate but most hardware only provides one and two qubit native gates decomposing the toffoli into these primitives is therefore essential a well known decomposition uses 6 cnot gates along with h t and t t dagger t gates requiring no ancilla qubits from qiskit import quantumcircuit toffoli decomposition into h t t cnot control qubits 0 1 target qubit 2 qc quantumcircuit 3 qc h 2 qc cx 1 2 qc tdg 2 qc cx 0 2 qc t 2 qc cx 1 2 qc tdg 2 qc cx 0 2 qc t 1 qc t 2 qc h 2 qc cx 0 1 qc t 0 qc tdg 1 qc cx 0 1 print qc draw this decomposition matters enormously for fault tolerant quantum computing in surface code error correction clifford gates h s cnot can be implemented transversally at relatively low cost t gates however require a resource called magic state distillation which is expensive each t gate costs roughly 10 to 100 times more than a clifford gate in terms of physical qubits and time the toffoli decomposition above uses 7 t t t dagger t gates so a single toffoli is quite expensive in a fault tolerant setting reducing the t count of circuits is an active area of research gate noise in practice real quantum gates are imperfect to understand the impact of noise on a computation you can simulate gate errors using qiskit aer s noise model from qiskit import quantumcircuit from qiskit_aer import aersimulator from qiskit_aer noise import noisemodel depolarizing_error build a bell state circuit qc quantumcircuit 2 2 qc h 0 qc cx 0 1 qc measure 0 1 0 1 create a noise model with realistic error rates noise_model noisemodel single qubit depolarizing error 0 1 error rate error_1q depolarizing_error 0 001 1 noise_model add_all_qubit_quantum_error error_1q h x y z s t two qubit depolarizing error 1 error rate error_2q depolarizing_error 0 01 2 noise_model add_all_qubit_quantum_error error_2q cx run with noise simulator aersimulator noise_model noise_model result simulator run qc shots 10000 result counts result get_counts print noisy bell state results counts ideal result 00 5000 11 5000 noisy result typical 00 4925 11 4925 01 75 10 75 in the ideal case a bell state produces only 00 and 11 outcomes with noise you see leakage into 01 and 10 the contamination rate tells you the effective circuit error if about 1 5 of shots produce wrong outcomes your total circuit error is approximately 1 5 this matches what you would expect from the error budget one h gate with 0 1 error and one cnot with 1 error gives roughly 1 1 total error with the remainder from depolarization spreading errors across multiple outcomes understanding this error budget is critical for deciding whether a quantum circuit is feasible on current hardware if your circuit has 100 cnot gates at 1 error each the total error is approximately 1 0 99 100 63 1 0 99 100 approx 63 1 0 99 100 63 making the output unreliable parameterized gates in variational algorithms the rotation gates rx ry and rz are the workhorses of variational quantum algorithms where classical optimization adjusts gate parameters to minimize a cost function vqe style ansatz in the variational quantum eigensolver vqe a typical ansatz uses ry gates for single qubit rotations and cnot for entanglement here is a 2 qubit example with 4 parameters import numpy as np from qiskit import quantumcircuit def vqe_ansatz params create a 2 qubit vqe ansatz with 4 parameters qc quantumcircuit 2 layer 1 single qubit rotations qc ry params 0 0 qc ry params 1 1 entangling layer qc cx 0 1 layer 2 single qubit rotations qc ry params 2 0 qc ry params 3 1 return qc example circuit with specific angles params 0 3 0 7 1 2 0 5 qc vqe_ansatz params print qc draw qaoa gate structure in the quantum approximate optimization algorithm qaoa two types of parameterized layers alternate the cost layer applies rzz gates implemented as cnot rz cnot and the mixer layer applies rx gates import numpy as np from qiskit import quantumcircuit def qaoa_layer gamma beta n_qubits 2 one layer of qaoa for a simple 2 qubit problem qc quantumcircuit n_qubits cost layer rzz gamma on each pair rzz gamma cnot i rz gamma cnot qc cx 0 1 qc rz gamma 1 qc cx 0 1 mixer layer rx beta on each qubit qc rx beta 0 qc rx beta 1 return qc qc qaoa_layer gamma 0 5 beta 0 3 print qc draw the parameter shift rule to optimize variational circuits you need gradients the parameter shift rule provides exact gradients not numerical approximations for rotation gates for any rotation gate r θ r theta r θ appearing in a circuit the partial derivative of the expectation value e langle e rangle e with respect to θ theta θ is e θ k e θ k π 2 e θ k π 2 2 frac partial langle e rangle partial theta_k frac langle e theta_k pi 2 rangle langle e theta_k pi 2 rangle 2 θ k e 2 e θ k π 2 e θ k π 2 this requires two circuit evaluations per parameter for a circuit with p p p parameters computing the full gradient requires 2 p 2p 2 p circuit evaluations import numpy as np def parameter_shift_gradient cost_fn params param_index compute the gradient of cost_fn with respect to params param_index using the parameter shift rule shift np pi 2 evaluate at theta pi 2 params_plus params copy params_plus param_index shift cost_plus cost_fn params_plus evaluate at theta pi 2 params_minus params copy params_minus param_index shift cost_minus cost_fn params_minus return cost_plus cost_minus 2 0 example compute gradient for all 4 parameters params np array 0 3 0 7 1 2 0 5 in practice cost_fn would run the vqe circuit and measure the expectation value of a hamiltonian the parameter shift rule works because rotation gates have exactly two eigenvalues 1 2 pm 1 2 1 2 in their generator this is a mathematical property specific to quantum gates and has no classical analogue native gate transpilation when you write a circuit using abstract gates like h t or s the quantum compiler must translate these into the hardware s native gate set on ibm quantum processors the single qubit natives are rz sx x where sx is the square root of x a 90 degree rotation around the x axis the two qubit native depends on the processor generation older falcon devices exposed cx directly eagle devices used ecr and the current heron devices use cz we use the cx based set below because it is the simplest to read here is how common gates decompose into this native set h gate rz π 2 sx rz π 2 text rz pi 2 cdot text sx cdot text rz pi 2 rz π 2 sx rz π 2 t gate rz π 4 text rz pi 4 rz π 4 s gate rz π 2 text rz pi 2 rz π 2 y gate x rz π text x cdot text rz pi x rz π or equivalently sx composed with rz gates from qiskit import quantumcircuit from qiskit compiler import transpile original circuit with abstract gates qc quantumcircuit 2 qc h 0 qc t 0 qc s 1 qc cx 0 1 qc h 1 print original circuit print qc draw print f original gate count qc size transpile to ibm s native gate set transpiled transpile qc basis_gates rz sx x cx optimization_level 2 print n transpiled circuit print transpiled draw print f transpiled gate count transpiled size count each gate type gate_counts transpiled count_ops print f gate breakdown dict gate_counts when you run this you will see that the 5 abstract gates expand into a larger number of native gates the h gates each become three native gates rz sx rz while t and s gates each become a single rz the transpiler also applies optimizations adjacent rz gates are merged and redundant gates are cancelled this is why native gate count not abstract gate count determines the true cost of a circuit a circuit that looks simple in terms of h and t gates may expand significantly after transpilation common mistakes to avoid 1 confusing cx and cz cx cnot flips the target qubit when the control is 1 1 rangle 1 cz applies a phase of 1 1 1 when both qubits are 1 1 rangle 1 they are related by hadamard gates on the target cx i h cz i h text cx i otimes h cdot text cz cdot i otimes h cx i h cz i h using one where you need the other produces wrong results without any error message because both are valid two qubit gates if your entangling operation is producing unexpected phases instead of bit flips or vice versa check whether you used cx when you meant cz 2 using degrees instead of radians all qiskit rotation gates use radians not degrees writing qc rx 90 0 does not rotate by 90 degrees it rotates by 90 radians which is approximately 14 3 full rotations the result is effectively a random rotation always use np pi 2 for a 90 degree rotation np pi for 180 degrees and so on import numpy as np from qiskit import quantumcircuit wrong this rotates by 90 radians not 90 degrees qc_wrong quantumcircuit 1 qc_wrong rx 90 0 correct 90 degree rotation around x axis qc_right quantumcircuit 1 qc_right rx np pi 2 0 3 treating global phase as significant the states ψ psi rangle ψ and e i ϕ ψ e i phi psi rangle e i ϕ ψ are physically identical quantum states they produce the same measurement probabilities and the same expectation values for all observables code that compares statevectors using or np array_equal will often fail because different implementations may produce the same state with different global phases instead compare states using fidelity ψ ϕ 2 1 langle psi phi rangle 2 1 ψ ϕ 2 1 if the states are identical up to global phase import numpy as np these represent the same quantum state state1 np array 1 0 0 state2 np array 1 0 0 global phase of pi state3 np array 1 j 0 i 0 global phase of pi 2 direct comparison fails print np array_equal state1 state2 false but they are the same state fidelity comparison works fidelity abs np dot state1 conj state2 2 print f fidelity fidelity 1 0 correctly identifies them as identical 4 forgetting that measurement is not a gate measurement collapses the quantum state and cannot be undone you cannot apply a gate after measurement to undo the collapse in qiskit operations placed after a measurement on the same qubit may produce unpredictable results or errors depending on the backend always place all quantum gates before the measurement barrier if you need to apply conditional operations based on measurement results use classical feedforward mid circuit measurement which is a distinct operation that explicitly conditions subsequent gates on classical bit values from qiskit import quantumcircuit wrong gate after measurement has no defined quantum behavior qc_wrong quantumcircuit 1 1 qc_wrong h 0 qc_wrong measure 0 0 qc_wrong x 0 this does not undo the measurement correct all gates before measurement qc_right quantumcircuit 1 1 qc_right h 0 qc_right x 0 qc_right measure 0 0 5 misidentifying clifford vs non clifford gates the t gate is not a clifford gate this distinction matters for two reasons first the gottesman knill theorem says circuits with only clifford gates are efficiently simulable classically including even a single t gate in a non trivial way can push the circuit beyond classical simulability second in fault tolerant computing t gates are far more expensive than clifford gates due to the need for magic state distillation if you are running a classical simulation for testing purposes and only need clifford operations verify that your circuit contains no t gates adding a t gate accidentally can cause your simulation to slow down dramatically when using stabilizer based simulators from qiskit import quantumcircuit qc quantumcircuit 3 qc h 0 qc cx 0 1 qc s 1 qc cx 1 2 check for non clifford gates clifford_gates h x y z s sdg cx cz swap id barrier non_clifford gate operation name gate qubits for gate in qc data if gate operation name not in clifford_gates if non_clifford print f warning circuit contains non clifford gates non_clifford else print circuit is purely clifford classically simulable via stabilizer formalism where to go next two combinations are worth committing to memory h cnot creates entanglement and h t cnot gives universal quantum computing every quantum algorithm from shor s factoring to variational eigensolvers is built from these fundamental pieces from here you can explore quantum error correction which shows how to protect computations from the gate noise we discussed above you can also study variational algorithms to see parameterized gates in action on real optimization problems try it yourself quantum circuit builder open full screen drop in hadamard pauli s t and cnot gates and watch the state vector and measurement probabilities update instantly it runs entirely in your browser no signup or install written by dr donovan who writes on quantum computing research hardware and industry at quantum zeitgeist dr donovan ran his first quantum circuit on ibm s 5 qubit quantum experience in 2018 and has not put the subject down since he built quantumcomputingcourses com because the material out there forced a choice nobody should have to make pop science hand waving at one end phd level physics with no on ramp at the other and tutorial code that had usually stopped working by the time you found it he also writes on quantum computing at quantum zeitgeist quantumzeitgeist com was this tutorial helpful yes no share ready to go deeper browse structured courses from coursera 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