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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):
1 rangle rightarrow 1 rangle 1 1 this operation does not change the measurement probabilities directly rather it affects interference patterns matrix 1 0 0 1 circuit symbol z y gate y combines the effects of x and z with a complex phase it appears in pauli decompositions error correction and the relationship y i x z y ixz y i x z matrix 0 i i 0 the hadamard gate h the hadamard gate is a fundamental gate in quantum computing it creates superposition placing a qubit into an equal mixture of 0 0 rangle 0 and 1 1 rangle 1 0 0 1 2 equal superposition 1 0 1 2 superposition with phase matrix 1 1 1 2 1 1 circuit symbol h why it matters without hadamard or equivalent gates quantum algorithms could not explore multiple states through interference every quantum algorithm that claims quantum parallelism relies on hadamard or similar gates to prepare superposition states a common misconception deserves correction here superposition does not mean the qubit tries all answers simultaneously rather it means the qubit exists in a linear combination of basis states and gates manipulate the amplitudes of those states so that correct answers interfere constructively while wrong answers interfere destructively the h gate is its own inverse apply h twice and you get back where you started h h i h cdot h i h h i identity this property makes h an involution if you need to undo a hadamard just apply another one qiskit put qubit in superposition then measure from qiskit import quantumcircuit qc quantumcircuit 1 1 qc h 0 0 0 1 2 qc measure 0 0 will give 0 or 1 with equal probability the cnot gate controlled not the cnot or cx gate operates on two qubits a control and a target if the control qubit is 1 1 rangle 1 it flips the target qubit if the control is 0 0 rangle 0 nothing happens truth table 00 00 control 0 target unchanged 01 01 control 0 target unchanged 10 11 control 1 target flipped 11 10 control 1 target flipped circuit symbol control target creating entanglement with h cnot the sequence h followed by cnot is perhaps the most well known two gate operation in quantum computing this procedure creates a bell state which is a maximally entangled pair of qubits qiskit create bell state from qiskit import quantumcircuit qc quantumcircuit 2 2 qc h 0 hadamard on qubit 0 superposition qc cx 0 1 cnot entangle qubit 0 qubit 1 qc measure 0 1 0 1 result always 00 or 11 never 01 or 10 this two gate circuit forms the basis for quantum teleportation superdense coding and quantum key distribution phase gates s and t phase gates rotate the qubit around the z axis of the bloch sphere while they do not change measurement probabilities in the standard basis they affect interference the mechanism that gives quantum algorithms their speedup the s gate phase gate applies a 90 degree phase rotation to 1 1 rangle 1 two s gates are equivalent to one z gate s 2 z s 2 z s 2 z matrix 1 0 0 i t gate pi 8 gate this gate applies a 45 degree phase rotation the t gate and its inverse t t dagger t is essential for universal quantum computing having just h and t gates plus cnot allows us to approximate any quantum operation to arbitrary precision via the solovay kitaev theorem matrix 1 0 0 e iπ 4 qiskit from qiskit import quantumcircuit qc quantumcircuit 1 qc s 0 s gate qc t 0 t gate qc tdg 0 t dagger inverse t gate the toffoli gate ccnot the toffoli gate features two control qubits and one target the target qubit flips only when both control qubits are in the 1 1 rangle 1 state this gate acts as a reversible and gate it computes the and of the two control bits into the target because any classical circuit can be built from and and not gates and the toffoli can simulate both reversibly a not is achieved by setting both controls to 1 1 rangle 1 it is universal for classical reversible computation circuit symbol control control target the toffoli gate appears in quantum arithmetic circuits quantum adders error correction and algorithms that need to implement classical boolean logic within a quantum circuit qiskit from qiskit import quantumcircuit qc quantumcircuit 3 qc ccx 0 1 2 toffoli flip qubit 2 if qubits 0 and 1 are 1 the swap gate swap exchanges two qubit states a b b a ab rangle rightarrow ba rangle ab ba it is used in quantum circuits when qubit connectivity is limited and you need to route qubits past each other circuit symbol a swap gate can be decomposed into three cnot gates qiskit from qiskit import quantumcircuit qc quantumcircuit 2 qc swap 0 1 direct swap or equivalently qc2 quantumcircuit 2 qc2 cx 0 1 qc2 cx 1 0 qc2 cx 0 1 rotation gates rx ry rz for fine grained control you can rotate a qubit by any angle θ theta θ around any axis of the bloch sphere these appear constantly in variational quantum algorithms like vqe and qaoa import numpy as np from qiskit import quantumcircuit qc quantumcircuit 1 qc rx np pi 4 0 rotate 45 around x axis qc ry np pi 3 0 rotate 60 around y axis qc rz np pi 2 0 rotate 90 around z axis same as s gate up to global phase the rotation gate matrices are r x θ cos θ 2 i sin θ 2 i sin θ 2 cos θ 2 r_x theta begin pmatrix cos theta 2 i sin theta 2 i sin theta 2 cos theta 2 end pmatrix r x θ cos θ 2 i sin θ 2 i sin θ 2 cos θ 2 r y θ cos θ 2 sin θ 2 sin θ 2 cos θ 2 r_y theta begin pmatrix cos theta 2 sin theta 2 sin theta 2 cos theta 2 end pmatrix r y θ cos θ 2 sin θ 2 sin θ 2 cos θ 2 r z θ e i θ 2 0 0 e i θ 2 r_z theta begin pmatrix e i theta 2 0 0 e i theta 2 end pmatrix r z θ e i θ 2 0 0 e i θ 2 any single qubit unitary can be written as u e i α r z β r y γ r z δ u e i alpha r_z beta r_y gamma r_z delta u e i α r z β r y γ r z δ for some choice of angles this is the zyz euler decomposition and it proves that ry rz is a universal set for single qubit gates gate sets and universality you do not need to implement every possible gate in quantum hardware a universal gate set is a small collection of gates that can approximate any quantum operation gate set notes h t cnot standard universal set for fault tolerant qc rx ry rz cnot common for variational algorithms rz sx x ecr or cz ibm quantum s native sets ecr on eagle cz on heron phasedxz cz or sqrt iswap google s superconducting processors native gate set hardware providers have native gate sets when you compile a circuit your gates are decomposed into only the gates the hardware can actually execute we explore this process in the transpilation section below how gates are physically implemented understanding what happens at the hardware level helps explain why some gates are harder than others two dominant hardware platforms implement gates in fundamentally different ways superconducting qubits ibm google rigetti a superconducting qubit is a tiny circuit containing a josephson junction cooled to about 15 millikelvin it behaves like an artificial atom with two energy levels separated by a frequency of roughly 5 ghz to apply a gate the control electronics send a shaped microwave pulse at the qubit s resonant frequency the qubit absorbs the pulse and its state rotates on the bloch sphere the rotation angle depends on the pulse amplitude and duration a pi pulse an x gate drives a full rabi oscillation from 0 0 rangle 0 to 1 1 rangle 1 on ibm hardware a typical single qubit gate takes about 20 to 200 nanoseconds depending on the gate type and calibration the pulse envelope shape matters a simple square pulse causes the qubit to leak population into higher energy levels outside the computational space gaussian shaped pulses reduce this leakage the drag derivative removal by adiabatic gate pulse shape adds a correction term proportional to the time derivative of the gaussian envelope suppressing leakage to the third energy level this is why pulse level programming frameworks like qiskit pulse exist precise control of the pulse shape directly affects gate fidelity two qubit gates on superconducting hardware rely on coupling between neighboring qubits typically through a shared microwave resonator or a tunable coupler an echoed cross resonance ecr pulse implements a cnot equivalent operation in roughly 300 to 600 nanoseconds on ibm s eagle generation hardware ibm s newer heron chips use tunable couplers and cz gates that are several times faster google s superconducting chips also use tunable couplers implementing two qubit gates in tens of nanoseconds trapped ions ionq quantinuum in a trapped ion processor individual atomic ions commonly ytterbium 171 or barium 133 are held in place by electromagnetic fields inside a vacuum chamber two internal electronic states of each ion serve as 0 0 rangle 0 and 1 1 rangle 1 single qubit gates are implemented by shining a focused laser beam onto an individual ion the laser drives stimulated raman transitions between the two qubit states a pi pulse on a trapped ion system typically takes 5 to 20 microseconds roughly 1000 times slower than superconducting gates two qubit gates use the shared motion phonon modes of the ion chain the molmer sorensen gate or the light shift gate entangles two ions by coupling their internal states through the collective vibrational mode this process typically takes 100 to 200 microseconds the speed coherence trade off superconducting gates are about 1000 times faster than trapped ion gates however trapped ion qubits have coherence times of seconds to minutes while superconducting qubits typically maintain coherence for 100 to 300 microseconds the ratio of coherence time to gate time the number of gates you can perform before decoherence dominates is comparable between the two platforms typically allowing hundreds to low thousands of sequential gates property superconducting trapped ion single qubit gate time 20 200 ns 5 20 μs two qubit gate time 100 500 ns 100 200 μs t1 coherence time 100 300 μs 1 10 s t2 coherence time 50 200 μs 0 5 5 s connectivity nearest neighbor all to all gate fidelity and error rates no physical gate operation is perfect gate fidelity quantifies how close the actual operation is to the ideal unitary for a target unitary u u u and the actual operation v v v the average gate fidelity is f tr u v 2 d d 2 d f frac text tr u dagger v 2 d d 2 d f d 2 d tr u v 2 d where d d d is the hilbert space dimension d 2 d 2 d 2 for single qubit d 4 d 4 d 4 for two qubit in practice fidelities are measured using randomized benchmarking which provides a robust estimate that is insensitive to state preparation and measurement errors a fidelity of 99 9 means a 0 1 error rate per gate this sounds small but a circuit with 1000 gates accumulates roughly 1 0 999 1000 63 1 0 999 1000 approx 63 1 0 999 1000 63 total error probability making the output nearly useless without error correction fidelity comparison across platforms 2024 benchmarks platform single qubit fidelity two qubit fidelity ibm eagle heron superconducting 99 9 99 0 99 5 google sycamore superconducting 99 9 99 5 ionq forte trapped ion 99 97 99 5 quantinuum h2 trapped ion 99 99 99 8 99 9 why are two qubit gates harder a two qubit gate requires coupling two distinct physical oscillators or two ions through a shared phonon bus this coupling introduces additional decoherence pathways energy exchange with the environment crosstalk with neighboring qubits and frequency collisions each additional coupling mechanism adds a potential error source this is a fundamental reason why quantum circuit optimization focuses heavily on minimizing two qubit gate count the controlled z cz gate the cz gate applies a phase of 1 1 1 when both qubits are in 1 1 rangle 1 and acts as the identity otherwise cz matrix 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 that is diag 1 1 1 1 relationship to cnot cz and cnot are closely related you can convert between them using hadamard gates on the target qubit cnot i h cz i h text cnot i otimes h cdot text cz cdot i otimes h cnot i h cz i h this means apply h to the target then cz then h to the target again and you get a cnot an important property of cz that cnot lacks is symmetry there is no designated control or target qubit cz 01 cz 10 text cz _ 01 text cz _ 10 cz 01 cz 10 either qubit can be called the control this symmetry makes cz a natural choice for hardware where the coupling between qubits is symmetric google s superconducting processors ibm s heron chips rigetti s chips and several other platforms natively implement cz rather than cnot every cnot in your circuit is compiled into cz plus two hadamard gates when targeting these backends verify the cz to cnot identity numerically import numpy as np define the gates h np array 1 1 1 1 np sqrt 2 i np eye 2 cz np diag 1 1 1 1 cnot from cz i h cz i h ih np kron i h cnot_from_cz ih cz ih standard cnot matrix cnot np array 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 print cnot from cz equals standard cnot np allclose cnot_from_cz cnot output true the iswap gate the iswap gate is native on some superconducting platforms particularly those that use a resonant coupling scheme between qubits its matrix is iswap matrix 1 0 0 0 0 0 i 0 0 i 0 0 0 0 0 1 the iswap gate swaps 01 01 rangle 01 and 10 10 rangle 10 while applying a phase factor of i i i iswap 01 i 10 text iswap 01 rangle i 10 rangle iswap 01 i 10 iswap 10 i 01 text iswap 10 rangle i 01 rangle iswap 10 i 01 the states 00 00 rangle 00 and 11 11 rangle 11 are left unchanged physically the iswap arises naturally when two superconducting qubits with the same frequency are coupled the excitation swaps between them with a phase accumulation this natural origin is why some platforms including google s earlier processors use iswap as a native gate to build a cnot from iswap gates you need two iswap gates plus single qubit rotations import numpy as np iswap matrix iswap np array 1 0 0 0 0 0 1 j 0 0 1 j 0 0 0 0 0 1 verify action on 01 state_01 np array 0 1 0 0 result iswap state_01 print 01 after iswap result output 0 0 j 0 0 j 0 1 j 0 0 j this is i 10 confirming the swap with phase i circuit identity proofs understanding circuit identities is essential for both hand optimization and understanding how compilers simplify quantum circuits here are several important identities with numerical verification identity 1 hxh z conjugating x by hadamard gates produces z intuitively h rotates the bloch sphere so that the x axis maps to the z axis identity 2 hzh x the reverse also holds conjugating z by hadamard gives x identity 3 xz iy up to global phase the product of x and z gates equals i y iy iy identity 4 cnot h h cz h h wrapping cz with hadamards on both qubits also produces a cnot with swapped control and target compared to the single hadamard identity above import numpy as np define basic gates x np array 0 1 1 0 y np array 0 1 j 1 j 0 z np array 1 0 0 1 h np array 1 1 1 1 np sqrt 2 i np eye 2 identity 1 h x h z result1 h x h print hxh z np allclose result1 z true identity 2 h z h x result2 h z h print hzh x np allclose result2 x true identity 3 x z iy result3 x z print xz iy np allclose result3 1 j y true identity 4 h h cz h h gives a cnot with swapped control target cz np diag 1...