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rx ry and rz reaching any point on the bloch sphere skip to main content quantumcomputing courses com courses all courses course platforms coursera edx udemy brilliant hardware providers google quantum ai ibm quantum ionq quantinuum amazon braket azure quantum quera rigetti d wave tutorials all tutorials hello world qiskit hello world cirq hello world pennylane hello world braket quantum gates grover s algorithm shor s algorithm reference all frameworks qiskit cirq pennylane amazon braket pyquil tket d wave ocean q explore learn learning paths prerequisites programming guide case studies glossary books quantum news podcasts tools bloch sphere quantum pinball guides algorithm guide hardware guide qubit types framework comparison migration guide language timeline cheat sheets career events 2026 jobs careers certifications salary guide universities interview prep faq troubleshooting about about team search browse courses home tutorials rx ry and rz reaching any point on the bloch sphere concepts intermediate free part 4 of 5 15 min read 7 jul 2026 by dr donovan quantum zeitgeist editorial policy rx ry and rz reaching any point on the bloch sphere give a rotation a dial instead of a fixed angle and you can steer a qubit anywhere on the bloch sphere with two gates this is the euler decomposition and it is what your transpiler does to every circuit you run bloch sphere rotation gates euler angles transpilation universality the bloch sphere hands on 1 how to read the bloch sphere 2 pauli gates as rotations x y and z on the bloch sphere 3 hadamard s and t on the bloch sphere 4 rx ry and rz reaching any point on the bloch sphere you are here 5 measurement decoherence and what the bloch sphere cannot show you prerequisites parts 1 to 3 of this series comfort with sine and cosine basic python for the optional code checks in this guide 6 sections 01 a rotation you choose 02 why the angle is halved 03 two rotations reach everything 04 what your transpiler is really doing 05 interrogating the decomposition yourself 06 where the sphere runs out every gate so far has been a rotation with the angle welded shut 180 for the paulis and h 90 for s 45 for t the rotation gates rx ry and rz take the weld off and give you a dial set the angle then pick an axis every gate in part 2 plus s and t is a preset on this panel h is the exception its axis is the diagonal which rx ry and rz cannot express on their own that one change is the difference between a handful of fixed moves and complete control of the qubit this part shows why two dialled rotations are enough to reach any state on the sphere and why that fact is the reason your circuits look nothing like what you wrote after the compiler has finished with them a rotation you choose set the slider to 45 and press ry ry 45 in flight same green y axis as the y gate in part 2 but the arrow stops a quarter of the way instead of going all the way over ry 45 leaves the qubit at θ 45 an 85 4 14 6 split not a coin flip a loaded one and loaded to a value you chose this is the first gate in the series that can produce a biased qubit h gives you 50 50 whether you like it or not ry 45 gives you 85 4 14 6 because p 0 cos² 45 2 cos² 22 5 0 854 any bias you want is one slider away which is what state preparation actually is rx behaves the same way about the red axis rx 90 in flight turning about the same red axis the x gate used it will stop halfway down at i rather than carrying on to the south pole turn the dial to 180 and you get the pauli gates back rx 180 does to the sphere exactly what x does strictly the matrices differ by a factor of i but that is a global phase and part 2 showed why the sphere is entitled to ignore it the paulis are the ends of these dials not separate machinery why the angle is halved ry 45 moved the arrow 45 on the sphere and the simulator s θ readout says 45 so where did the half angle of part 1 go it is in the matrix not the picture the rotation gate is defined as ry α exp i α y 2 cos α 2 i i sin α 2 y the α 2 inside the gate and the θ 2 inside the state cancel out exactly which is the whole reason the bloch picture is worth having the angle you dial is the angle the arrow sweeps underneath the state vector in hilbert space only turned half as far a 360 rotation brings the arrow home but leaves the state at ψ and only a 720 turn restores the state itself that is the notorious spin 1 2 double cover and on the sphere you get to enjoy the benefit while ignoring the bookkeeping two rotations reach everything here is the payoff a state is just two angles θ and φ you have a gate that sets each of them ry θ tips the arrow down from the north pole by θ that fixes the latitude rz φ spins it around by φ that fixes the longitude so starting from 0 ry then rz reaches any point on the sphere try it set the slider to 70 press ry set it to 130 press rz the second rotation rz 130 sweeping the arrow around at constant latitude θ 70 φ 130 exactly as dialled the panel even writes out the state cos 35 0 e i130 sin 35 1 the readout gives back precisely the numbers you asked for there was no searching no optimisation no cleverness latitude then longitude and any state on the sphere is two gates away that is universality for a single qubit and it is why hardware vendors do not need to give you a hundred gates any single qubit unitary whatsoever can be written as three rotations u rz γ ry β rz α up to a global phase three angles because a general unitary also has to say what it does to every state not just to 0 these are the euler angles the same ones used to orient an aircraft for the same reason both problems are describe an arbitrary rotation of a sphere what your transpiler is really doing this is not decorative theory it is the core of transpilation real hardware does not implement h or t or the gate you actually wrote a superconducting processor typically offers a native gate set of roughly rz sx a 90 x rotation and x plus one two qubit gate every single qubit gate in your circuit gets rewritten into that alphabet and the euler decomposition is the tool that does the rewriting there is a lovely piece of hardware trickery hiding here on superconducting qubits rz is free it is not executed as a physical pulse at all the control electronics simply relabel the phase reference for every subsequent pulse a virtual z zero duration zero error rotations about x and y need real microwave pulses which take time and add noise so the decomposition above is not merely one valid choice among many it is the economical one pushing as much of the work as possible onto the free axis watch qiskit do it import numpy as np from qiskit import quantumcircuit transpile from qiskit quantum_info import statevector the state from the screenshots theta 70 phi 130 qc quantumcircuit 1 qc ry np radians 70 0 qc rz np radians 130 0 sv statevector qc p0 abs sv data 0 2 print f p 0 p0 100 1f p 1 1 p0 100 1f print f cos 2 35 deg np cos np radians 35 2 100 1f p 0 67 1 p 1 32 9 cos 2 35 deg 67 1 the 67 1 32 9 split is exactly what the simulator s probability bars showed now hand the same circuit to the transpiler and ask for hardware native gates native transpile qc basis_gates rz sx x optimization_level 3 print native print gate counts dict native count_ops global phase π 2 q x rz 1 9199 x rz 0 87266 gate counts sx 2 rz 2 your ry and rz came back as four gates and that looks like a loss until you count what will actually cost you anything the two rz gates are free so the whole state preparation costs two physical pulses the two x the transpiler has rebuilt your ry out of the free rotation and the one real pulse the hardware owns sandwiching phase shifts around fixed 90 x pulses this is also why the gate count printed by a transpiler is a bad proxy for how hard a circuit is to run ask instead how many pulses and how many t gates interrogating the decomposition yourself qiskit will hand you the euler angles for any single qubit unitary directly from qiskit quantum_info import operator from qiskit synthesis import onequbiteulerdecomposer decomposer onequbiteulerdecomposer basis zyz for name in h x s t theta phi lam _ decomposer angles_and_phase operator from_label name data print f name rz np degrees lam 7 1f ry np degrees theta 7 1f rz np degrees phi 7 1f h rz 180 0 ry 90 0 rz 0 0 x rz 0 0 ry 180 0 rz 180 0 s rz 90 0 ry 0 0 rz 0 0 t rz 45 0 ry 0 0 rz 0 0 the gates are listed in the order they would run left to right read the table and every claim in this series is confirmed at a glance s and t have no ry component at all they are bare z rotations 90 and 45 which is exactly why part 3 found them doing nothing whatsoever to states parked on the z axis x is a 180 y rotation with a z shuffle attached there is more than one route to the south pole and the decomposer picked a different one from the x axis which is fine any of them produce the same unitary h is rz 180 followed by ry 90 feed 0 into that and the rz does nothing 0 is on its axis then the 90 tip drops it onto the equator at the famous superposition gate is a phase flip you cannot see followed by a quarter turn you can where the sphere runs out you can now put a single qubit anywhere you like with two gates and you know what the compiler will turn them into which is the natural moment to ask what this picture cannot do it has no way to draw an entangled pair it has no way to draw a qubit that is decohering and it has said nothing yet about what happens at the only moment that matters when you measure part 5 is about the edges of the map try it yourself interactive bloch sphere open full screen apply x y z h s t and rotation gates and watch the qubit state rotate in 3d in real time it runs entirely in your browser no signup or install previous part 3 hadamard s and t on the bloch sphere next part 5 measurement decoherence and what the bloch sphere cannot show you 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 edx udemy brilliant and more browse courses related tutorials continue learning with these guides bell inequalities and the chsh test how we know entanglement is real intermediate 30 min read read the bernstein vazirani algorithm implementing a hidden string finder in qiskit beginner 20 min read read hadamard s and t on the bloch sphere beginner 14 min read read previous turing machines computability and what quantum computing actually changes next measurement decoherence and what the bloch sphere cannot show you on this page 01 a rotation you choose 02 why the angle is halved 03 two rotations reach everything 04 what your transpiler is really doing 05 interrogating the decomposition yourself 06 where the sphere runs out at a glance level intermediate read time 15 min read language python updated jul 2026 related tutorials bell inequalities and the chsh test how we know entanglement is real 30 min read the bernstein vazirani algorithm implementing a hidden string finder in qiskit 20 min read hadamard s and t on the bloch sphere 14 min read courses on this quantum computing 101 from zero to your first circuit free quantum computing 201 algorithms noise and real hardware free get one quantum email a week new tutorials courses worth taking and what changed in qiskit cirq pennylane this week no spam unsubscribe anytime email address subscribe 112 courses 220 tutorials 241 glossary terms 26 framework 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