Chapter 9. Conditions for Quantum Hardware

So far, we have treated qubits, gates, algorithms, and error correction codes as if they operate perfectly on paper. However, implementing all this theoretical framework in reality is one of humanity’s greatest engineering challenges.

Building quantum hardware is a battle against a fundamental dilemma. Quantum computers must be completely isolated from the surrounding environment to maintain coherence. But at the same time, to perform algorithms, we must precisely control and measure qubits. This is inherently an act of intervening in the system.

In this chapter, we learn about the key requirements that physical systems must meet to overcome this dilemma and build useful quantum computers, namely DiVincenzo’s Criteria and the key measures of coherence, T1, T2 times.

1. Fundamental Concepts

DiVincenzo’s 5 Criteria: Five essential requirements for a useful quantum computer, proposed by David DiVincenzo in 2000. These have become the standard benchmark for evaluating all quantum hardware platforms.
1. Scalability: The system must be scalable to accommodate millions of well-defined qubits.
2. Initialization: All qubits must be reset to a reliable initial state such as \(|00\dots0\rangle\). (Starting point for algorithms)
3. Long Coherence Times: The quantum state of qubits must be maintained for much longer than the gate operation time (\(\tau_{gate}\)). (\(T_{coherence} \gg \tau_{gate}\))
4. Universal Control: The universal gates learned in Chapter 1 (e.g., single-qubit rotation + CNOT) must be precisely applied to qubits.
5. Measurement: The final state of individual qubits must be measurable with high accuracy.
T1 Time (Energy Relaxation Time): A measure of the rate of energy loss in qubits. (Longitudinal Relaxation)
- Definition: The average time it takes for a qubit in the excited state (\(|1\rangle\)) to spontaneously lose energy and decay to the ground state (\(|0\rangle\)).
- Error Model: Directly linked to the Amplitude Damping channel in Chapter 6.
- Significance: T1 determines how long a qubit can retain information in the \(|1\rangle\) state, i.e., the ‘maximum time limit’ for computations.
T2 Time (Transverse Relaxation Time): A measure of the rate of phase information loss in qubits. (Transverse Relaxation)
Definition: When a qubit is in a superposition state such as \(|+\rangle = \frac{|0\rangle+|1\rangle}{\sqrt{2}}\), it is the average time it takes for the delicate relative phase relationship between \(|0\rangle\) and \(|1\rangle\) to be randomized and collapse.
Error Model: It is directly related to the Phase-Flip channel in Chapter 6.
Meaning: T2 indicates how long a qubit’s ‘superposition’ can be maintained. If T2 is zero, the qubit behaves like a classical bit.
Relationship between T1 and T2:
T2 is always shorter than T1. (Energy loss (\(T_1\)) automatically causes phase loss (\(T_2\)), but phase loss (\(T_2\)) can occur without energy loss (\(T_1\))).
\[T_2 \le 2T_1\]
Generally, \(T_2\) is much shorter than \(T_1\), and \(T_2\) practically limits the performance of quantum algorithms.
Quantum Volume (Quantum Volume, QV):
The number of qubits alone cannot describe a computer’s performance. The quality of qubits (error rate) is also important. Quantum Volume is a modern integrated benchmark that indicates “how many qubits and how deep (complex) a circuit can be successfully executed.” If QV is \(2^L\), it means that \(L\) qubits can reliably execute a circuit of depth \(L\).

2. Symbols and Key Relations

T1 (Energy Relaxation):
The probability \(P_1(t)\) of being in the \(|1\rangle\) state decays exponentially.
\[P_1(t) = P_1(0) e^{-t/T_1}\]
T2 (Phase Decoherence):
The off-diagonal coherence component \(\rho_{01}\) of the density matrix in a superposition state decays exponentially.
\[\rho_{01}(t) = \rho_{01}(0) e^{-t/T_2}\]
T2* (T2-star) Time:
The measured T2 time arises from two causes.
1. Irreversible: “Real” decoherence due to entanglement with the environment.
2. Reversible: Slow phase changes caused by static magnetic field noise unique to each qubit. (e.g., reversible with spin-echo techniques)
- \(\frac{1}{T_2} = \frac{1}{T_{2, \text{irreversible}}} + \frac{1}{T_{2, \text{static}}}\) (T2* is shorter than \(T_2\))
Gate Fidelity \(F\):
A measure of how accurately a gate performs the intended unitary operation \(U\) (\(\rho \to \mathcal{E}(\rho)\)).
\[F = \langle\psi_{ideal}| \mathcal{E}(|\psi_{ideal}\rangle\langle\psi_{ideal}|) |\psi_{ideal}\rangle\]
- Error Rate (\(\epsilon\)): \(\epsilon = 1 - F\)
Performance Criterion (Criterion 3):
If an algorithm requires \(N_{gates}\) gates, then
\[T_2 \gg N_{gates} \times \tau_{gate}\]
or, in terms of gate error rate \(\epsilon\), \(N_{gates} \times \epsilon \ll 1\) must hold.

3. Simplified Examples (Examples with Deeper Insight)

Example 1: T1 - Energy Damping (Amplitude Damping)

Situation: The qubit is prepared in the \(|1\rangle\) state. This qubit undergoes the ‘amplitude damping channel’ learned in Chapter 6. (e.g., an excited atom emits a photon and falls into the ground state)
- Physical Meaning:
  The collapse from \(|1\rangle \to |0\rangle\) is irreversible energy loss.
  If \(T_1 = 50 \mu\text{s}\) (microseconds), then after \(50 \mu\text{s}\), the probability that the qubit remains in the \(|1\rangle\) state is only \(e^{-1} \approx 37\%\). After \(100 \mu\text{s}\), it is \(\approx 13.5\%\).
- 💡 Detailed Explanation:
  > T1 error causes bit-flip (\(X\)) error in the Z-basis (\(|0\rangle, |1\rangle\)). \(T_1\) is a limit on how long classical information (0 or 1) stored in the qubit can be maintained.

Example 2: T2 - Dephasing (Phase Damping)

Situation: The qubit is prepared in the \(|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\) state.
Physical Meaning:
The \(|+\rangle\) state oscillates at a frequency (\(\omega = \Delta E / \hbar\)) corresponding to the energy difference \(\Delta E\) between \(|0\rangle\) and \(|1\rangle\). \(\psi(t) = \frac{1}{\sqrt{2}}(|0\rangle + e^{-i\omega t}|1\rangle)\).

However, environmental noise (e.g., tiny magnetic fields caused by neighboring spins) slightly perturbs this \(\Delta E\). (\(\omega \to \omega + \delta\omega(t)\))

Over time, this random phase shift \(\delta\omega(t)\) accumulates, causing the qubit to completely lose the initial phase relationship between \(|0\rangle\) and \(|1\rangle\).
💡 Detailed Explanation (T1 vs T2 - Spinning Top Analogy)

The qubit’s state (Bloch vector) can be metaphorically compared to a spinning top.
- Direction of the Top: A top standing upright along the Z-axis (\(|0\rangle\)) represents the \(|0\rangle\) state. A top lying horizontally and rotating in the XY-plane (\(|+\rangle\)) represents the \(|+\rangle\) state.
- T1 (Energy Loss): The process where the top loses speed and falls over due to friction. (e.g., \(|+\rangle \to |0\rangle\) or \(|1\rangle \to |0\rangle\))
- T2 (Phase Loss): The top does not fall over (no \(T_1\) loss), but its axis of rotation (Z-axis) slightly wobbles, causing the rotation speed (phase) of the top to become irregular.
Key Point: It may take a long time (\(T_1\)) for the top to fall over, but making the top’s rotation phase unpredictable happens much more easily and quickly (\(T_2\)). This is why \(T_2 \ll T_1\), and why quantum algorithms (which rely on superposition) are far more sensitive to \(T_2\).

Example 3: Gate Count Calculation (Criterion 3)

Situation: Current state-of-the-art superconducting qubits have a coherence time of \(T_2 = 150 \mu\text{s}\) (0.00015 seconds). Performing one CNOT gate takes \(\tau_{gate} = 50 \text{ ns}\) (0.00000005 seconds).
Problem: How many CNOT operations can be performed with this qubit before coherence is lost?
Calculation:
Maximum number of gates \(\approx T_2 / \tau_{gate} = (150 \times 10^{-6} \text{ s}) / (50 \times 10^{-9} \text{ s})\)
\(= 3 \times 10^3 = 3,000\)
💡 Detailed Explanation (Necessity of Error Correction)
> 3,000 seems large, but the Shor algorithm for factoring \(N=2048\) bits requires about \(10^9\) (one billion) gates.
> Even running 3,000 gates on hardware where information is lost is impossible. This shows why Quantum Error Correction Code (QECC), learned in Chapters 7 and 8, is not just a theoretical concept but a necessary condition for building a fault-tolerant quantum computer. We must perform thousands of error “correction” cycles before a single computation finishes.

4. Practice Problems

Deutsch-Jozsa Criterion: List the five criteria of Deutsch-Jozsa that are essential for algorithm execution.
T1 and T2: If a qubit in the \(|1\rangle\) state experiences a \(T_1\) error (energy relaxation), what state does it become? If a qubit in the \(|+\rangle\) state experiences a \(T_2\) error (dephasing), what state does it ideally become? (Hint: Chapter 6)
Limitations of T2: A qubit’s \(T_1\) time is measured as \(100 \mu\text{s}\). Can its \(T_2\) time be \(300 \mu\text{s}\)? Why or why not?
Gate Error Rate: A qubit has \(T_2 = 200 \mu\text{s}\) and gate time \(\tau_{gate} = 100 \text{ ns}\). Approximately calculate the coherence-limited error rate \(\epsilon \approx \tau_{gate} / T_2\) for this gate.
Initialization: Explain why Deutsch-Jozsa’s second criterion (initialization) is important, using the first step of the Deutsch-Jozsa algorithm from Chapter 2 as an example.

5. Explanation

1. Scalable qubit, (2) Initialization, (3) Long coherence time, (4) Universal gate, (5) Measurement.
T1 error: \(|1\rangle\) state becomes \(|0\rangle\) state after undergoing \(T_1\) process. (Energy loss) T2 error: \(|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\) state becomes mixed state \(\rho = \frac{1}{2}(|0\rangle\langle 0| + |1\rangle\langle 1|) = \frac{\mathbf{1}}{2}\) after undergoing \(T_2\) process, with the off-diagonal terms vanishing. (Quantum superposition collapse)
None. According to the relation \(T_2 \le 2T_1\), the \(T_2\) of a qubit with \(T_1 = 100 \mu\text{s}\) can be at most \(200 \mu\text{s}\). \(T_2\) includes not only energy relaxation (T1) but also pure dephasing process, so it must be shorter than (or equal to) \(T_1\).
\(\epsilon \approx \tau_{gate} / T_2 = (100 \times 10^{-9} \text{ s}) / (200 \times 10^{-6} \text{ s}) = 0.5 \times 10^{-3} = 0.0005\). This implies that the error rate per gate is approximately \(0.05\%\) (or fidelity 99.95%) due only to the coherence time. (In reality, the error rate is higher due to gate control inaccuracies, etc.)
The first step of the Deutsch-Jozsa algorithm (Chapter 2) is to apply \(H^{\otimes n}\) to the \(|0\rangle^{\otimes n}\) state to create a uniform superposition (\(\sum |x\rangle\)) of all inputs. If the qubit starts in a random state (e.g., \(|1\rangle\) mixed in due to T1 error, or a less measured state) rather than \(|0\rangle\), applying \(H^{\otimes n}\) will not result in a uniform superposition, and the algorithm’s interference pattern will be corrupted, leading to incorrect answers.