Quantum Computation (Caltech PHYS 219)
Prof. John Preskill, Caltech
A qubit is the basic unit of quantum information. It behaves nothing like a classical bit. Understanding what a qubit actually is - mathematically, physically, and practically - is the foundation of everything else in quantum computing.
A classical bit is always in one of two states: 0 or 1. A qubit is described by a state vector in a two-dimensional complex vector space. Written in Dirac notation, the general state of a qubit is:
|ψ⟩ = α|0⟩ + β|1⟩ Here α and β are complex numbers (amplitudes) satisfying |α|² + |β|² = 1. The values |α|² and |β|² give the probabilities of measuring 0 or 1 respectively.
This is not the same as saying the qubit is "both 0 and 1 at the same time." Superposition means the qubit genuinely has no definite value before measurement - it is in a quantum state that is neither 0 nor 1, but a combination that collapses to a specific outcome only when observed. The act of measurement itself forces the qubit into a definite state, and the superposition is destroyed.
This is why you cannot simply read out the amplitudes α and β directly. Each measurement gives only one bit of classical information: 0 or 1. Extracting useful information from quantum states requires careful circuit design and often many repeated measurements.
A qubit is an abstract mathematical object. To build one in hardware, engineers need a two-level quantum system whose states can be prepared, manipulated with gates, and measured with high reliability. Several physical approaches are in active use today.
IBM, Google, Rigetti
Superconducting qubits use tiny circuits of superconducting material cooled to temperatures near absolute zero (around 15 millikelvin). A non-linear inductor called a Josephson junction creates an anharmonic energy level structure, making it possible to isolate the lowest two energy levels as |0⟩ and |1⟩. Gate operations are applied with microwave pulses. They are fast (gate times in nanoseconds) and manufactured with semiconductor fabrication tools, but require extreme cooling and are sensitive to noise from their environment.
IonQ, Quantinuum
Individual ions are trapped using electromagnetic fields and laser-cooled to near rest. The internal electronic energy levels of the ion serve as |0⟩ and |1⟩. Gate operations use laser pulses or microwave radiation. Trapped-ion qubits have some of the highest gate fidelities and longest coherence times of any platform, and any two qubits in the trap can interact directly. The main challenges are gate speed (microseconds rather than nanoseconds) and scaling to large numbers of ions in a single trap.
PsiQuantum, Xanadu
Photonic qubits encode quantum information in properties of single photons such as polarization or path. They can be transmitted over optical fiber and do not require cryogenic cooling, making them attractive for quantum networking and room-temperature operation. The fundamental challenge is that photons do not interact naturally with each other, making two-qubit gates difficult. Measurement-based approaches and specialized nonlinear elements are used to work around this limitation.
QuEra, Pasqual, Atom Computing
Individual neutral atoms are held in place by tightly focused laser beams called optical tweezers. Qubit states are encoded in the hyperfine levels of the atom's ground state. Two-qubit gates exploit Rydberg interactions - when one atom is excited to a high-energy Rydberg state, it blockades neighboring atoms from being excited simultaneously, providing a controlled interaction. Neutral atoms offer long coherence times, reconfigurable connectivity, and the ability to arrange large 2D arrays of qubits.
A superconducting quantum processor inside its dilution refrigerator. The gold structure keeps qubits cooled to around 15 millikelvin.
When evaluating a quantum processor or comparing hardware platforms, four properties of qubits matter most in practice.
T1 measures how long a qubit initialized in the excited state |1⟩ takes to spontaneously decay back to |0⟩. This sets a hard upper limit on how long any computation can run. A T1 of 100 microseconds means you have roughly 100 microseconds of useful operation before the qubit has a significant chance of spontaneously flipping. Modern superconducting qubits typically achieve T1 values between 100 and 500 microseconds; trapped ions can reach seconds.
T2 measures how long a qubit maintains phase coherence in a superposition state. Dephasing is a subtler form of error than energy relaxation: the qubit's amplitude does not change, but the relative phase between |0⟩ and |1⟩ drifts randomly due to environmental noise. T2 is always less than or equal to 2*T1. For most algorithms, T2 is the more constraining timescale because many operations depend on phase relationships between qubits.
Gate fidelity is the probability that a gate operation produces the exact state it was intended to produce, averaged over all possible input states. A two-qubit gate fidelity of 99.5% sounds good, but compounding 100 such gates gives an overall success probability below 60%. High-fidelity gates are essential for running circuits with many operations, and they are harder to achieve for two-qubit gates than single-qubit gates. State-of-the-art systems reach 99.9% or above for single-qubit gates and 99-99.5% for two-qubit gates.
Not every qubit can interact directly with every other qubit on most hardware. Superconducting processors typically connect each qubit to 2-5 neighbors in a fixed graph; trapped-ion systems offer all-to-all connectivity within the trap. When an algorithm requires a two-qubit gate between non-adjacent qubits, the compiler must insert SWAP gates to route the interaction through intermediate qubits. Extra SWAP gates add circuit depth, increase error, and consume coherence time, so hardware topology directly affects which algorithms run efficiently on a given machine.
Courses that teach qubit theory, quantum information, and the mathematical foundations of quantum computing.
Prof. John Preskill, Caltech
DAMTP, University of Cambridge
Delft University of Technology (QuTech)
Delft University of Technology (QuTech)
MIT xPRO
Austin Fowler
Prof. Isaac Chuang and Prof. Peter Shor, MIT
Prof. Isaac Chuang and Prof. Peter Shor, MIT