# The Qubit > The qubit is the hydrogen atom of quantum computing — the simplest nontrivial quantum system, and > the one we'll never stop using. Here we make it concrete: what it is physically, how superposition > and the Born rule look in $\mathbb{C}^2$, and — for the first time in this program — how to **build > and run a real quantum circuit on AWS Braket**. Everything stays on the free local simulator. ## Learning Objectives After this lesson you will be able to: 1. Define a **qubit** as a two-level quantum system and give physical realizations. 2. Write a general qubit state and compute computational-basis **measurement probabilities**. 3. Explain **superposition** and how a single Hadamard creates it. 4. Build, run, and read out a circuit on the Braket **`LocalSimulator`** (free). 5. Connect the simulator's amplitudes and shot counts to the Born rule. --- ## Intuition A bit is a switch: $0$ or $1$. A **qubit** is a quantum two-level system whose state can be any *superposition* of $\lvert0\rangle$ and $\lvert1\rangle$ — a complex blend that, when measured, yields $0$ or $1$ with probabilities set by the amplitudes. Physically a qubit can be the spin of an electron ($\uparrow/\downarrow$), the polarization of a photon (H/V), two energy levels of an atom or a superconducting circuit (Term 4.4), and more. What makes all of them "qubits" is shared mathematics: the state space is $\mathbb{C}^2$, and the rules are the four postulates of Course 1.1. --- ## Theory ### The qubit state By Postulate 1 ([1.1.1](../c01-postulates/01-state-postulate.md)), a qubit's state is a unit vector in $\mathbb{C}^2$: $$ \lvert\psi\rangle = \alpha\lvert0\rangle + \beta\lvert1\rangle, \qquad \alpha,\beta\in\mathbb{C}, \qquad |\alpha|^2 + |\beta|^2 = 1, $$ with the computational basis $\lvert0\rangle = \begin{psmallmatrix}1\\0\end{psmallmatrix}$, $\lvert1\rangle = \begin{psmallmatrix}0\\1\end{psmallmatrix}$. The amplitudes $\alpha,\beta$ are complex; their magnitudes give probabilities and their *relative* phase encodes interference ([1.1.1](../c01-postulates/01-state-postulate.md)). ### Born rule for a qubit Measuring in the computational basis ($Z$ observable, [1.1.2](../c01-postulates/02-observables-measurement-postulate.md)): $$ p(0) = |\langle0|\psi\rangle|^2 = |\alpha|^2, \qquad p(1) = |\langle1|\psi\rangle|^2 = |\beta|^2, \qquad p(0)+p(1)=1. $$ A result of $0$ collapses the state to $\lvert0\rangle$; a result of $1$ to $\lvert1\rangle$. ### Superposition and the Hadamard The most important single-qubit operation for *creating* superposition is the **Hadamard** gate $H = \tfrac1{\sqrt2}\begin{psmallmatrix}1&1\\1&-1\end{psmallmatrix}$ ([Appendix E](../../appendix/E-math-identities.md)): $$ H\lvert0\rangle = \tfrac1{\sqrt2}(\lvert0\rangle + \lvert1\rangle) = \lvert+\rangle, \qquad H\lvert1\rangle = \tfrac1{\sqrt2}(\lvert0\rangle - \lvert1\rangle) = \lvert-\rangle. $$ Starting from the definite state $\lvert0\rangle$, one $H$ produces an equal superposition: measuring it in the computational basis gives $0$ or $1$ with probability $\tfrac12$ each. This is the "fair quantum coin," and it's the natural first circuit to run. (Gates get the full treatment in Term 2; here we use $H$ as our entry point.) --- ## Worked Examples ### Example 1 — Probabilities of a general qubit For $\lvert\psi\rangle = \tfrac1{\sqrt3}\lvert0\rangle + \sqrt{\tfrac23}\,e^{i\pi/4}\lvert1\rangle$: $p(0) = |1/\sqrt3|^2 = 1/3$ and $p(1) = |\sqrt{2/3}\,e^{i\pi/4}|^2 = 2/3$ (the phase $e^{i\pi/4}$ does not affect the magnitude). Sum $= 1$. ✓ The phase *would* matter if we measured in the $X$ or $Y$ basis ([1.2.3](03-bases-and-expectation.md)). ### Example 2 — The Hadamard "coin" Apply $H$ to $\lvert0\rangle$: $\lvert+\rangle = \tfrac1{\sqrt2}(\lvert0\rangle+\lvert1\rangle)$, so a $Z$-measurement gives $0$ and $1$ each with probability $\tfrac12$. Apply a *second* $H$: since $H^2 = I$, $HH\lvert0\rangle = \lvert0\rangle$ — measuring now gives $0$ with certainty. Two "random" coin flips compose to a deterministic result: **interference**, not classical randomness. We'll see this directly on the simulator below. --- ## Hands-on (Python) ### First, in NumPy ```python import numpy as np ket0 = np.array([1, 0], dtype=complex) ket1 = np.array([0, 1], dtype=complex) H = np.array([[1, 1], [1, -1]], dtype=complex) / np.sqrt(2) plus = H @ ket0 # |+> print(np.round(plus, 4)) # [0.7071 0.7071] print(np.abs(plus)**2) # [0.5 0.5] Born probabilities print(np.allclose(H @ (H @ ket0), ket0)) # True: HH = I (interference -> back to |0>) ``` ### Now on AWS Braket (local simulator — free) ⚙️ This is your first Braket circuit. It runs entirely on your machine via `LocalSimulator` — **no AWS account, no cost** (see [Appendix A](../../appendix/A-braket-setup.md)). Three concepts: a `Circuit` holds gates; `LocalSimulator` executes it; `shots` is how many times we sample the measurement. ```python # the_qubit.py — free, local, no AWS account needed. from braket.circuits import Circuit from braket.devices import LocalSimulator # Build a one-qubit circuit: H on qubit 0 prepares |+>. circ = Circuit().h(0) print(circ) # ASCII circuit diagram # Run it. Braket measures all qubits in the computational (Z) basis by default. device = LocalSimulator() # default backend: state-vector ("braket_sv") result = device.run(circ, shots=1000).result() print(result.measurement_counts) # ~ Counter({'0': ~500, '1': ~500}) ``` ```text T : |0| q0 : -H- Counter({'0': 508, '1': 492}) ``` The counts are *samples* from the Born distribution $p(0)=p(1)=\tfrac12$ — finite-shot estimates with the $1/\sqrt{M}$ error from [0.2.2](../../term-0-foundations/c02-probability-information/02-random-variables-expectation.md). Now confirm the interference of Example 2 — two Hadamards return a *definite* $0$: ```python circ2 = Circuit().h(0).h(0) # H then H = identity on |0> counts = LocalSimulator().run(circ2, shots=1000).result().measurement_counts print(counts) # Counter({'0': 1000}) — always 0 ``` If you also want the exact amplitudes (simulator-only superpower; real hardware never gives these), add a state-vector result type: ```python circ3 = Circuit().h(0) circ3.state_vector() # request the full amplitude vector sv = LocalSimulator().run(circ3, shots=0).result().values[0] print(np.round(sv, 4)) # [0.7071+0.j 0.7071+0.j] = |+> ``` > ⚠️ **Cost reminder.** `LocalSimulator` is free. The on-demand simulators (SV1/DM1/TN1) and QPUs are > billed — we don't touch them until Term 5, and never automatically. Everything in Terms 1–4 runs > locally. --- ## Exercises **E1 (easy).** For $\lvert\psi\rangle = \sqrt{0.2}\,\lvert0\rangle + \sqrt{0.8}\,\lvert1\rangle$, give the $Z$-measurement probabilities and predict the approximate counts for 1000 shots.

Solution

$p(0)=0.2$, $p(1)=0.8$. Over 1000 shots expect ≈ 200 `'0'` and ≈ 800 `'1'`, with statistical spread $\sim\sqrt{1000\cdot0.2\cdot0.8}\approx 13$ counts.

**E2 (easy).** Write the Braket circuit that prepares $\lvert1\rangle$ from the default $\lvert0\rangle$, and predict its measurement counts.

Solution

`Circuit().x(0)` (Pauli-$X$ = NOT flips $\lvert0\rangle\to\lvert1\rangle$). Counts: all `'1'` (`Counter({'1': 1000})`), since the state is the definite $\lvert1\rangle$.

**E3 (medium).** Using only NumPy, verify that applying $H$ to $\lvert1\rangle$ gives $\lvert-\rangle$ and that its $Z$-measurement probabilities are also $\tfrac12,\tfrac12$. Why do $\lvert+\rangle$ and $\lvert-\rangle$ give identical computational-basis statistics despite being different states?

Solution

$H\lvert1\rangle = \tfrac1{\sqrt2}(\lvert0\rangle-\lvert1\rangle)=\lvert-\rangle$; $|{\pm}1/\sqrt2|^2=\tfrac12$ each. They share $Z$-statistics because those depend only on amplitude *magnitudes*; the distinguishing relative phase ($+$ vs $-$) only shows up in a different basis (e.g. $X$), where $\lvert+\rangle\to0$ and $\lvert-\rangle\to1$ with certainty — see [1.2.3](03-bases-and-expectation.md).

**E4 (medium).** Run (or simulate by hand) `Circuit().h(0).h(0).h(0)` on $\lvert0\rangle$. What are the output statistics and why?

Solution

Three Hadamards: $H^3 = H^2\cdot H = I\cdot H = H$, so the state is $\lvert+\rangle$ and the counts are ≈ 50/50. An odd number of $H$'s acts like a single $H$; an even number like the identity.

**E5 (hard).** A state $\lvert\psi\rangle = \cos\tfrac\theta2\lvert0\rangle + \sin\tfrac\theta2\lvert1\rangle$ (real amplitudes). You run many shots and observe $p(1) \approx 0.15$. Estimate $\theta$, and state how many shots you'd need to pin $p(1)$ to within $\pm0.01$ at 95% confidence.

Solution

$p(1) = \sin^2(\theta/2) = 0.15 \Rightarrow \sin(\theta/2) = \sqrt{0.15}\approx0.387 \Rightarrow \theta/2\approx0.398 \Rightarrow \theta\approx0.80$ rad ($\approx46°$). For a $\pm\varepsilon$ estimate of a probability at 95% confidence, Hoeffding gives $M \ge \frac{1}{2\varepsilon^2}\ln\frac{2}{\delta}$ with $\varepsilon=0.01$, $\delta=0.05$: $M \ge \frac{1}{2(0.01)^2}\ln(40) \approx \frac{\ln 40}{2\times10^{-4}}\approx 1.84\times10^4$ shots (from [0.2.2](../../term-0-foundations/c02-probability-information/02-random-variables-expectation.md)).

--- ## Checkpoint 1. What is a qubit, and what is its state space? Name two physical realizations. 2. Write a general qubit state and its computational-basis measurement probabilities. 3. What does a single Hadamard do to $\lvert0\rangle$, and what do *two* do? Why? 4. In the Braket code, what are `Circuit`, `LocalSimulator`, `shots`, and `measurement_counts`? 5. Why are local-simulator runs free, and which Braket resources are not?

Answers

1. A two-level quantum system; state space $\mathbb{C}^2$. Realizations: electron spin, photon polarization, atomic/superconducting energy levels (any two). 2. $\lvert\psi\rangle=\alpha\lvert0\rangle+\beta\lvert1\rangle$ with $|\alpha|^2+|\beta|^2=1$; $p(0)=|\alpha|^2$, $p(1)=|\beta|^2$. 3. One $H$: $\lvert0\rangle\to\lvert+\rangle$ (equal superposition). Two: $H^2=I$ returns $\lvert0\rangle$ — interference, not random. 4. `Circuit` is the gate sequence; `LocalSimulator` runs it on your machine; `shots` = number of measurement samples; `measurement_counts` is the histogram of outcome bit-strings. 5. `LocalSimulator` runs on your own CPU (no AWS service invoked); on-demand simulators (SV1/DM1/TN1) and QPUs are billed cloud resources (Term 5).

--- ## Further Reading - **[NC]** Nielsen & Chuang, §1.2, §2.2 — the qubit and superposition. - **[Mer]** Mermin, *Quantum Computer Science*, Ch. 1 — qubits for computer scientists. - **[SDK]** `amazon-braket-sdk-python` README & examples — the `Circuit` / `LocalSimulator` API. --- ← Prev: [1.1.4 Composite Systems Postulate](../c01-postulates/04-composite-systems-postulate.md) · Up: [Term 1](../README.md) · Next: [The Bloch Sphere](02-bloch-sphere.md) →