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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):
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 1 1 1 hh np kron h h result4 hh cz hh this equals cnot with qubit 1 as control qubit 0 as target cnot_10 np array 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 print h h cz h h cnot_10 np allclose result4 cnot_10 true these identities are how quantum circuit compilers simplify gate sequences for example if a compiler sees h x h in sequence it can replace it with a single z gate reducing the circuit depth by two thirds multi qubit controlled gates the cnot gate is a special case of a general controlled u gate where u is any single qubit unitary the controlled u gate applies u to the target qubit only when the control qubit is 1 1 rangle 1 the matrix form is c u 0 0 i 1 1 u c u 0 rangle langle 0 otimes i 1 rangle langle 1 otimes u c u 0 0 i 1 1 u this means if the control is 0 0 rangle 0 do nothing if the control is 1 1 rangle 1 apply u u u important special cases include controlled z cz u z u z u z the gate we discussed above controlled s cs u s u s u s applies a 90 degree phase when both qubits are 1 1 rangle 1 controlled t ct u t u t u t applies a 45 degree phase when both qubits are 1 1 rangle 1 decomposing controlled s into native gates any controlled phase gate can be decomposed into cnot gates plus single qubit rotations for the controlled s gate from qiskit import quantumcircuit import numpy as np build controlled s from cnot and rz gates qc quantumcircuit 2 qc rz np pi 4 1 rz pi 4 on target qc cx 0 1 cnot qc rz np pi 4 1 rz pi 4 on target qc cx 0 1 cnot qc rz np pi 4 0 rz pi 4 on control for global phase correction print qc draw the significance of controlled t the controlled t gate holds a special position in fault tolerant quantum computing the gate set h s cnot generates the clifford group discussed below which is efficiently simulable by classical computers adding the t gate breaks out of the clifford group and together with the clifford gates generates a set that is dense in all unitaries the controlled t gate is the key ingredient that enables the full clifford t hierarchy which is the standard framework for fault tolerant quantum computation clifford gates and the clifford group the clifford group consists of all unitary operations that map pauli operators to pauli operators under conjugation more precisely a unitary c c c is clifford if for every pauli operator p p p the conjugation c p c c p c dagger c p c yields another pauli operator up to a phase of 1 pm 1 1 or i pm i i the clifford group is generated by three gates h s cnot all pauli gates x y z are also clifford gates since they are products of h and s here are the key conjugation relations that define how clifford generators transform pauli operators import numpy as np 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 s np array 1 0 0 1 j h conjugation h maps x z print h x h z np allclose h x h conj t z true print h z h x np allclose h z h conj t x true s conjugation s maps x y print s x s y np allclose s x s conj t y true print s z s 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 ...
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