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?

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.

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.

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!

Got it, here’s a tighter version that uses a simple Dirichlet‑series oracle. I’ll keep the depth low and add a few more counting qubits for resolution. ```python from qiskit import QuantumCircuit, Aer, execute from qiskit.circuit.library import QFT from qiskit.transpiler import PassManager, transpile from qiskit.providers.aer.noise import NoiseModel from qiskit.algorithms import PhaseEstimation from qiskit.utils import QuantumInstance from qiskit.algorithms.utils import get_custom_circuit from qiskit.circuit.library import CXGate # ----- 1. Dirichlet series oracle ----- def dirichlet_oracle(alpha, num_target): """Approximate exp(2πi * α * sum_{n=1}^N 1/n^s) using controlled rotations.""" qc = QuantumCircuit(num_target) N = 16 # number of terms to sum for n in range(1, N+1): coeff = 1 / (n ** 2) # example: s=2 angle = 2 * 3.1415926535 * alpha * coeff qc.rz(angle, 0) # rotate the first qubit as a proxy return qc # ----- 2. Build Phase Estimation circuit ----- num_count = 6 # counting qubits num_target = 3 # target qubits for Dirichlet oracle pe = PhaseEstimation(quantum_instance=QuantumInstance(Aer.get_backend('qasm_simulator'), shots=1024)) circuit = pe.construct_circuit( num_qubits=num_count, unitary=dirichlet_oracle(alpha=0.7, num_target=num_target), qubit_list=[i for i in range(num_count + num_target)] ) # ----- 3. Transpile for a real device ----- backend = Aer.get_backend('ibmq_qasm_simulator') # replace with actual backend circuit = transpile(circuit, backend, optimization_level=3, basis_gates=['u3', 'cx'], layout_method='trivial', routing_method='dense') # ----- 4. Readout error mitigation ----- # create a simple mitigation object from qiskit.algorithms.utils import get_pre_post_mitigator mitigator = get_pre_post_mitigator(backend) circuit = mitigator.apply(circuit) # ----- 5. Execute ----- job = execute(circuit, backend, shots=2048) result = job.result() counts = result.get_counts(circuit) print(counts) ``` **What to look for** - A peak in the histogram that shifts as you change `alpha`. - If the plot is flat, re‑run with different shot seeds and average. - The fridge can stay humming, just don’t let it drown the signal. Hope this gets us closer to a zero in the phase estimate!