Conditions for quantum computation

Physical Realizations of Qubits

A quantum computer must strike a balance between two conflicting needs:

1.  Isolation – to preserve quantum coherence.
2.  Accessibility – to allow operations and readout.

A quantum computer has to be well isolated... but its qubits have to be accessible...

These timescales determine how “good” a quantum system is:

1. $\tau_{Q}$: Coherence time: how long a qubit remains quantum.
2. $\tau_{op}$: Operation time: time to apply a gate.
3. $\lambda = \tau_{op} / \tau_{Q}$: Noise strength (smaller is better).
4. $n_{op} = \lambda^{-1}$: Max ops before decoherence.

$$\begin{array}{|l|c|c|c|} \hline \textbf{System} & \tau_Q & \tau_{op} & n_{op} = \lambda^{-1} \\ \hline \text{Nuclear spin} & 10^{-2} \text{ to } 10^{8} & 10^{-3} \text{ to } 10^{-6} & 10^5 \text{ to } 10^{14} \\ \text{Electron spin} & 10^{-3} & 10^{-7} & 10^4 \\ \text{Ion trap (In}^+) & 10^{-1} & 10^{-14} & 10^{13} \\ \text{Electron – Au} & 10^{-8} & 10^{-14} & 10^6 \\ \text{Electron – GaAs} & 10^{-10} & 10^{-13} & 10^3 \\ \text{Quantum dot} & 10^{-6} & 10^{-9} & 10^3 \\ \text{Optical cavity} & 10^{-5} & 10^{-14} & 10^9 \\ \text{Microwave cavity} & 10^{0} & 10^{-4} & 10^4 \\ \hline \end{array}$$

4 basic requirements

1. Robustly represent quantum infomation
2. Perform a universal family of unitary transformations
3. Perpare a fiducial initial state
4. Measure the output result

Representation of quantum infomation

Quantum computation is based on transforming quantum states. Qubits are two‐level systems that provide a convenient finite state space for computation.

1. Finite state space is crucial

   • Continuous variables (e.g. position $x$) inhabit an infinite‐dimensional Hilbert space—unrealistic once noise is included.
   • Noise limits the number of distinguishable states to a finite set.

   For example, in a perfect world, the entire texts of Shakespeare could be stored in the infinite number of digits in the binary fraction $x = 0.010111011001...$. What happens in reality is that the presence of noise reduces the number of distinguishable states to a finite number.

2. Symmetry enforces finiteness

   • It is generally desirable to have some aspect of symmetry dictate the finiteness of the state space, in order to minimize decoherence.
   • A spin-$\tfrac12$ particle lives in the two‐dimensional space spanned by $|\!\!\uparrow\rangle$ and $|\!\!\downarrow\rangle$. When well isolated, this is an almost ideal qubit.

3. Poor representations lead to decoherence

   • Example: a finite square well with exactly two bound states still couples to the continuum—transitions destroy superpositions.
   • Any leakage out of the two‐level subspace adds noise.

4. Figures of merit for single qubits

   • $T_2$ (transverse relaxation time): minimum lifetime of arbitrary superpositions (best measure of coherence).
   • $T_1$ (longitudinal relaxation time): lifetime of the energy eigenstate $|1\rangle$ (a “classical” lifetime, usually $>T_2$).

“Anything which causes loss of quantum information is a noise process.”

Performance of unitary transformations

Closed quantum systems evolve under their Hamiltonian, but quantum computation requires the ability to control that Hamiltonian to implement any desired gate.

1. Hamiltonian Control

   • Evolution under
     
     $$ H = P_x(t)\,X + P_y(t)\,Y $$

     where $P_{x,y}(t)$ are classical control parameters.

   • By shaping $P_x$ and $P_y$, one can perform arbitrary single‐spin rotations.

2. Universal Gate Set

   • Any unitary can be decomposed into single‐spin rotations + CNOT gates.
   • Requires addressability: Implicitly required also is the ability to address individual qubits, and to apply these gates to select qubits or pairs of qubits. e.g. in an ion trap you must focus a laser on individual ions (spaced $\geq$ one wavelength).

3. Imperfections → Decoherence

   • Unrecorded imperfections in unitary transfoms can lead to decoherence.
   • Random errors: uncontrolled “kicks” (small rotations about $\hat z$) introduce random relative phases → loss of coherence.
   • Systematic errors: calibration drifts accumulate into irreversible noise if you lose the information needed to reverse them.
   • Back‐action: controls are quantum too. For example, a Jaynes–Cummings interaction
         $$ P_x(t)=\sum_k \omega_k(t)\,(a_k + a_k^\dagger) $$     couples the qubit to the photon field, which can carry away state information.

4. Figures of Merit

   • Fidelity $\mathcal{F}$: minimum achievable fidelity of the target unitary.
   • Operation time $t_{op}$: maximum time needed for elementary gates (rotations, CNOT).

High‐precision control of $H$ and suppression of all error sources are key to achieving high‐fidelity quantum gates.

Preparation of fiducial initial states

Quantum computation requires a reliable method to prepare a known input state. This is non-trivial for quantum systems.

1. Classical vs Quantum Input

   • In classical computers, inputs are trivial (bit switches).
   • In quantum systems, preparing a known state (e.g. all spins in $|0\rangle$) is hard due to system-dependent constraints.

2. Sufficient Input

   • Only one known pure state is needed (e.g. $|00\ldots0\rangle$), since unitary evolution can generate any other state.
   • Challenge: maintaining the state due to heating or noise.

3. Physical Realization

   • Ions: prepared via laser cooling into their ground state.

   • Ensembles (e.g. NMR):

     • Each molecule is a qubit.
     • Many molecules needed for a measurable signal.
     • Hard to align all in the same quantum state.

   • In thermal equilibrium:
     
     $$ \rho \approx \frac{e^{-\mathcal{H}/k_B T}}{\mathcal{Z}} $$

     where $\mathcal{Z}$ normalizes $\text{tr}(\rho) = 1$.

4. Figures of Merit

   • Fidelity: accuracy of preparing a target state $\rho_{\text{in}}$.
   • Entropy of $\rho_{\text{in}}$:
     • Example: $\rho_{\text{in}} = I/2^n$ is easy to make, but useless (max entropy, fully mixed).
     • Ideal state = pure, zero entropy.

Good computation begins with a pure input. Noise in the input reduces information accessibility.

Measurement of output result

Quantum computation requires a way to extract classical results from quantum states.

1. Basic Measurement Model

   • Couple qubit to classical system → observe final state.
   • Example:
     • State $a|0\rangle + b|1\rangle$ measured via fluorescence:
       • Detect light → collapse to $|1\rangle$ with prob. $|b|^2$
       • No light → collapse to $|0\rangle$

2. Wavefunction Collapse

   • Projective measurement maps quantum superposition to classical value.
   • In algorithms like Shor's, output is superposition over $c$ values; collapse gives random $c$ to infer period $r$.

3. Measurement Challenges

   • Noise: Photon loss, amplifier thermal noise, inefficient detection.
   • Strong measurements: Require large, switchable coupling → technically hard and may introduce decoherence.
   • Timing: Measurement must not occur prematurely.

4. Weak & Ensemble Measurements

   • Weak measurements: always-on, continuous coupling can work.
   • Ensemble readouts: large groups of qubits give a macroscopic signal, e.g. NMR systems.
   • But: ensemble returns $\langle c \rangle$, not discrete $c$ → averaging can break algorithms needing exact integers.
   • Solution: modify algorithm for ensemble-compatible readout.

5. Figure of Merit

   • Signal-to-Noise Ratio (SNR): Measures how distinguishable the output is despite inefficiencies or weak signals.

Measurement must be precise, controllable, and not disrupt the quantum state until the computation is complete.

References

[1]. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Ed., Cambridge: Cambridge University Press, 2010.