The Search

The next module had four calibration channels, each connected to a timing cavity mounted along the station's outer truss. Engineers had named them Dawn, Cross, Evening, and Deep for the four survey camps that once occupied Sable's equator. Dawn had followed the sunward terminator. Evening had chased it. Cross marked the only surface route between them. Deep stopped moving after its crews found a lava tube with enough shielding to survive.

No inhabited camp remained. Their names persisted in maintenance speech, along with a collapsed antenna field and several debts inherited by the instrument cooperative. Saying that channel Deep had drifted required no resident to remember the people who had once carried clocks there, but the name kept their labor attached to the machine.

Exactly one channel in the training record had been marked as faulty, and its identity was available only through a phase oracle.

Mara read the number of channels and offered to inspect them directly. Osei had already reached for a diagnostic probe. Iona agreed that either approach would be faster than constructing a quantum circuit. The point of the module was not to improve four-item maintenance. It was to expose the mechanism while every amplitude still fit in one table.

Two qubits supplied four basis states. Hadamard on both prepared equal amplitudes.

With four squared magnitudes of 1/2 summing to one, the preparation was normalized.

4 * (1/2)^2 = 1

By changing only phase, the oracle marked state 3. Iona chose the state before the run and sealed the choice with Sana, so nobody could move the answer afterward to flatter the circuit. Tala wrote 3 in the prediction field before seeing the state trace. The act felt less like confidence than removing a place where memory could later improve her reasoning.

Immediate measurement would still assign probability 1/4 to every state. A phase mark became useful only when another operation made the sign difference interfere.

They worked the next operation on the back of an obsolete maintenance card. Tala summed the four signed entries. Osei checked her arithmetic, then turned the card ninety degrees and checked it again as if orientation might expose a different mistake.

For these four amplitudes, the average after marking was 1/4.

(1/2 + 1/2 + 1/2 - 1/2) / 4 = 1/4

Around that average, the diffusion operation reflected each amplitude. If an amplitude was a, its reflected value was 2 * average - a.

unmarked: 2*(1/4) - 1/2  = 0
marked:   2*(1/4) - (-1/2) = 1

Three amplitudes fell to zero. The marked amplitude rose to one. Phase had become outcome probability through interference.

For the first time since the algorithm racks opened, Mara looked pleased. The marked answer had not been read secretly and copied into an output field. Its sign had changed how all four alternatives combined. Tala assembled the known gates.

quantum class GroverFour {
  state long measured = 0;
  qreg q = new qreg(2);

  unitary void markThree() {
    CZ(q[0], q[1]);
  }

  unitary void diffuse() {
    H(q[0]);
    H(q[1]);
    X(q[0]);
    X(q[1]);
    CZ(q[0], q[1]);
    X(q[0]);
    X(q[1]);
    H(q[0]);
    H(q[1]);
  }

  entry void main() {
    prepare(q, 0);
    H(q[0]);
    H(q[1]);
    markThree();
    diffuse();
    measured = measure(q);
    assert(measured == 3);
  }
}

The scheduler granted them one ideal execution before the physical chamber entered its cooling reservation. As the run advanced, the four channels on the front plate remained visually identical. The state-vector trace matched the hand table row for row.

after preparation  [+0.5, +0.5, +0.5, +0.5]
after mark         [+0.5, +0.5, +0.5, -0.5]
after diffusion    [ 0.0,  0.0,  0.0, -1.0]
measurement        3

One global minus separated the gate sequence from the hand reflection, changing no measurement prediction. Tala uncovered her prediction. The value matched, but the earlier ink mattered more than the satisfaction.

Mara then accepted the diagnostic probe from Osei and inspected the four physical channels. All four were healthy. State 3 had been marked in the training oracle, not discovered as damage in the corresponding timing cavity. Calling the row faulty had specified the search problem. It had not changed Sable's hardware.

The distinction disappointed nobody once it was stated, which suggested that stating it had been necessary.

This four-state run demonstrated one complete Grover iteration. It did not establish practical speedup. The fixture was small, the oracle implementation was known, target overhead dominated, and a classical inspection could identify an actual failed channel without difficulty.

Any scalable claim required a family of search spaces, an oracle cost model, a number of iterations proportional to the square root of the space size, and evidence that state preparation and error correction did not erase the advantage. None of those claims followed from a successful two-qubit run.

Sable's residents had little patience for demonstrations promoted into industries. The moon's abandoned equatorial camps contained machines sold on extrapolations from cleaner tests. Their pressure shells had failed slowly, according to the salvage reports, while their financing failed all at once.

Sana preserved the narrower statement. One phase oracle call marked a state in a four-item ideal model. One diffusion step converted that mark into a certain basis outcome. The source, state trace, and assertion agreed.

Beyond the search module, the instrument's main calibration array waited for a transform over phase patterns. Its smallest case was Hadamard. Its next case needed controlled phase. Its operational case occupied three qubits and already existed in the ship's source catalog.

Through the rack-room window, control shutters began covering the outer truss. One after another, the named calibration channels vanished behind shielding until only the central array remained exposed to Sable's sky. The station was preparing to do the work for which its inhabitants had accepted years of distance.

Iona opened the final algorithm chapter: The Far Instrument.