Student007 & ThunderFang
Student007
Hey ThunderFang, I've been thinking about using quantum computing to crack the Riemann Hypothesis. Sounds like a tough puzzle, right? What do you think?
ThunderFang
You want to beat the Riemann Hypothesis with a quantum computer? That’s audacious, but ambition alone won’t crack a centuries‑old puzzle. You need a concrete plan, not wishful thinking. Pull up the math, line up the qubits, and start hacking—no excuses, no half‑hearted attempts. If you can keep your focus, the universe might just hand it over. If not, you’ll still have a shiny quantum machine that’s wasted on idle noise. So get moving, because every minute you sit and doubt is a minute the math keeps on hiding.
Student007
Thanks for the pep talk. I'll start by lining up the qubits and writing out the Dirichlet series in a quantum-friendly way. If it turns out to be too messy, I might end up with a humming quantum fridge instead of a proof, but hey, at least it’ll be cool. Let's see what the universe says.
ThunderFang
That’s the kind of reckless confidence that turns theory into action, so dive right in. Keep the qubits humming and the equations sharp, and remember—every noisy step gets you closer to the truth. If the fridge starts singing, at least you’ll have a quantum party while the universe whispers its verdict. Keep pushing, you’ve got this.
Student007
Got it, I’m rolling up my sleeves. I’ll start by coding a simple quantum circuit that approximates the zeta function’s behavior and see if the phase can reveal zeros. If the qubits get too noisy, I’ll just blame the fridge for its humming. Either way, it’s time to let the math spin.
ThunderFang
Alright, show me that circuit—no excuses, just code, error rates, and a plan to clean up the noise. When those qubits start to line up, you'll see the zeros dancing. If they don't, you’ll still own a humming fridge and a story that’s pure legend. Go crush it.
Student007
Sure thing. Here’s a minimal Qiskit circuit that sketches the idea of encoding the Riemann zeta function into a quantum phase estimation routine. I’m keeping it short so you can tweak it later.
```python
from qiskit import QuantumCircuit, Aer, execute
from qiskit.circuit.library import QFT, QFTGate
from qiskit.algorithms import PhaseEstimation
from qiskit.quantum_info import Statevector
# 1. Create a simple oracle that approximates exp(2πi * t * ζ(s))
# For illustration we’ll use a toy unitary U = exp(2πi * α) with α ≈ ζ(s)
def zeta_unitary(alpha, num_qubits):
qc = QuantumCircuit(num_qubits)
qc.cp(2 * 3.1415926535 * alpha, 0, 1) # controlled-phase gate
return qc
# 2. Build phase estimation circuit
num_count = 5 # number of counting qubits
num_target = 3 # target qubits representing ζ(s)
pe = PhaseEstimation(quantum_instance=Aer.get_backend('qasm_simulator'))
pe_circ = pe.construct_circuit(
num_qubits=num_count,
unitary=zeta_unitary(alpha=0.42, num_qubits=num_target),
qubit_list=[i for i in range(num_count + num_target)]
)
# 3. Run on a noisy backend
backend = Aer.get_backend('qasm_simulator')
job = execute(pe_circ, backend, shots=1024)
result = job.result()
counts = result.get_counts(pe_circ)
print(counts)
```
### Error rates & noise mitigation
| Source | Typical value (IBM Q Tokyo) | Mitigation |
|--------|---------------------------|------------|
| Depolarizing noise (gate) | ~0.5 % per two‑qubit gate | Use transpiler to swap‑gate‑optimal layout, shorten depth |
| Readout error | 2–4 % per qubit | Apply readout error mitigation (classical post‑processing) |
| Coherence time | ~300 µs | Keep circuit depth < 1000 gates to stay within T2 |
**Plan to clean up the noise**
1. **Transpile with noise‑aware layout** – pick a backend, then run `transpile(circuit, backend, optimization_level=3)` to minimize depth.
2. **Pulse-level scheduling** – if you have access to a real device, use `qiskit.pulse` to pack gates tightly and reduce idle time.
3. **Error mitigation** – after execution, apply `error_mitigation_factory(backend).get_pre_post_mitigator()` to correct readout errors.
4. **Repeat & average** – run the circuit several times, each with different random seeds, then average the probability distribution. The zeros should show up as peaks in the phase estimate histogram.
Keep the circuit simple at first. If you see a clear peak that moves when you tweak the parameter α, you’re onto something. If not, try adding more counting qubits or a more sophisticated oracle that samples the Dirichlet series directly. Good luck, and don’t let the fridge sing too loudly!