Mathematics · Quantum Mathematics
Bloch Sphere Qubit State Calculator
Convert Bloch-sphere angles into qubit amplitudes and measurement probabilities.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Convert angles: θ=1.5707963267948966 rad; φ=0 rad.
- α=cos(θ/2)=0.7071067811865476; β=sin(θ/2)e^(iφ)=0.7071067811865475+(0)i.
- Probabilities: P(0)=0.5000000000000001; P(1)=0.4999999999999999; total=1.
Understand Bloch sphere state
One idea, three depths
Choose how deeply to explain Bloch sphere state
Bloch sphere state: Convert Bloch-sphere angles into qubit amplitudes and measurement probabilities.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Bloch sphere state to answer this question: convert bloch-sphere angles into qubit amplitudes and measurement probabilities? Enter Polar angle θ and Azimuth φ; the calculator shows Probability P(0). For example: At θ=90° and φ=0°, the state is an equal superposition with P(0)=P(1)=0.5. The answer tells you Probability P(0).
Age 15Explain it to a 15-year-oldConnect it to the formula
Every pure qubit state, apart from an irrelevant global phase, corresponds to a point specified by polar angle θ and azimuth φ on the Bloch sphere. The rule is |ψ⟩=cos(θ/2)|0⟩+eⁱᵠsin(θ/2)|1⟩. Its input values are Polar angle θ (°), Azimuth φ (°), and the main result is Probability P(0). For example: At θ=90° and φ=0°, the state is an equal superposition with P(0)=P(1)=0.5.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated bloch sphere state relation over the valid real-number domain stated below. The implemented relation is |ψ⟩=cos(θ/2)|0⟩+eⁱᵠsin(θ/2)|1⟩, evaluated from Polar angle θ (°), Azimuth φ (°) to produce Probability P(0). Every pure qubit state, apart from an irrelevant global phase, corresponds to a point specified by polar angle θ and azimuth φ on the Bloch sphere. The amplitude uses half the polar angle θ, and φ changes relative phase rather than the measurement probabilities in the computational basis.
Inputs and valid domain
- Polar angle θ must be a finite real number in °.
- Azimuth φ must be a finite real number in °.
Important boundary: The amplitude uses half the polar angle θ, and φ changes relative phase rather than the measurement probabilities in the computational basis.
The formula
|ψ⟩=cos(θ/2)|0⟩+eⁱᵠsin(θ/2)|1⟩
How the calculator works through it
It substitutes Polar angle θ, Azimuth φ into the formula and exposes every numerical step above. The main output is Probability P(0), accompanied by Probability P(1), Amplitude α, β real component, β imaginary component.
Read the result correctly
The Probability P(0) is the direct answer to “convert bloch-sphere angles into qubit amplitudes and measurement probabilities.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
At θ=90° and φ=0°, the state is an equal superposition with P(0)=P(1)=0.5.
Where this model stops being reliable
The amplitude uses half the polar angle θ, and φ changes relative phase rather than the measurement probabilities in the computational basis.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Bloch sphere state works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Bloch sphere state uses |ψ⟩=cos(θ/2)|0⟩+eⁱᵠsin(θ/2)|1⟩. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Probability and normalised outcomes
Probability interpretation is needed to connect the Bloch sphere state mathematics to measurable outcomes.
Review this foundation about 6 min
Optional enrichment
- Complex amplitudes
Complex-number notation gives deeper context for amplitudes and phase relationships related to Bloch sphere state.
Review this foundation about 7 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Polar angle θ, Azimuth φ.
- Evaluate the principal relationship: |ψ⟩=cos(θ/2)|0⟩+eⁱᵠsin(θ/2)|1⟩.
- Return Probability P(0) and check the domain conditions described above.
Python
from math import *
def qubit_bloch_state(a, b) -> float:
return (cos((((a * pi) / 180.0) / 2.0)) * cos((((a * pi) / 180.0) / 2.0)))
assert abs(qubit_bloch_state(90, 0) - 0.5000000000000001) < 1e-6 * max(1.0, abs(0.5000000000000001))
C
#include <assert.h>
#include <math.h>
double qubit_bloch_state(double a, double b) {
return (cos((((a * 3.141592653589793) / 180.0) / 2.0)) * cos((((a * 3.141592653589793) / 180.0) / 2.0)));
}
int main(void) {
const double expected = 0.5000000000000001;
const double actual = qubit_bloch_state(90, 0);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double qubit_bloch_state(double a, double b) {
return (std::cos((((a * std::numbers::pi) / 180.0) / 2.0)) * std::cos((((a * std::numbers::pi) / 180.0) / 2.0)));
}
int main() {
constexpr double expected = 0.5000000000000001;
const double actual = qubit_bloch_state(90, 0);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double qubit_bloch_state(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern cos
global qubit_bloch_state
section .text
qubit_bloch_state:
push rbp
mov rbp, rsp
sub rsp, 144
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-64], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-64]
movsd [rbp-56], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-72], xmm0
movsd xmm0, [rbp-56]
divsd xmm0, [rbp-72]
movsd [rbp-48], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-80], xmm0
movsd xmm0, [rbp-48]
divsd xmm0, [rbp-80]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-40]
call cos wrt ..plt
movsd [rbp-32], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-120], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-120]
movsd [rbp-112], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-128], xmm0
movsd xmm0, [rbp-112]
divsd xmm0, [rbp-128]
movsd [rbp-104], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-136], xmm0
movsd xmm0, [rbp-104]
divsd xmm0, [rbp-136]
movsd [rbp-96], xmm0
movsd xmm0, [rbp-96]
call cos wrt ..plt
movsd [rbp-88], xmm0
movsd xmm0, [rbp-32]
mulsd xmm0, [rbp-88]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = qubit_bloch_state(a, b)
result = (cos((((a * pi) / 180.0) / 2.0)) * cos((((a * pi) / 180.0) / 2.0)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_] := (Cos[(((a * Pi) / 180.0) / 2.0)] * Cos[(((a * Pi) / 180.0) / 2.0)]);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
University Physics Volume 3
Read OpenStax University Physics: Quantum MechanicsCite this book
- APA 7
- Ling, S. J., Sanny, J., & Moebs, W. (2016). University physics volume 3. OpenStax. https://openstax.org/books/university-physics-volume-3/pages/1-introduction
- MLA 9
- Ling, Samuel J., et al. University Physics Volume 3. OpenStax, 2016, https://openstax.org/books/university-physics-volume-3/pages/1-introduction.
- Chicago author-date
- Ling, Samuel J., Jeff Sanny, and William Moebs. 2016. University Physics Volume 3. Houston, TX: OpenStax. https://openstax.org/books/university-physics-volume-3/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Bloch Sphere Qubit State Calculator. MW SysArc Tools. https://math.mwsysarc.com/quantum-mathematics/bloch-sphere-qubit-state
MLA 9
MW SysArc. “Bloch Sphere Qubit State Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/quantum-mathematics/bloch-sphere-qubit-state. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Bloch Sphere Qubit State Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/quantum-mathematics/bloch-sphere-qubit-state.
Harvard
MW SysArc (2026) ‘Bloch Sphere Qubit State Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/quantum-mathematics/bloch-sphere-qubit-state (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_qubit_bloch_state_2026,
author = {{MW SysArc}},
title = {Bloch Sphere Qubit State Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/quantum-mathematics/bloch-sphere-qubit-state},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Bloch Sphere Qubit State Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/quantum-mathematics/bloch-sphere-qubit-state
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Bloch sphere state do?
Convert Bloch-sphere angles into qubit amplitudes and measurement probabilities.
How does the Bloch sphere state work?
The calculator applies |ψ⟩=cos(θ/2)|0⟩+eⁱᵠsin(θ/2)|1⟩. Every pure qubit state, apart from an irrelevant global phase, corresponds to a point specified by polar angle θ and azimuth φ on the Bloch sphere.
What can I learn from the Bloch sphere state?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .