A GHZ state on qubits,

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Open quantum systems, decoherence and error correction · Open quantum systems, decoherence and error correction

Quantum Information and Technology, Ch The stabilizer formalism

Problem

A GHZ state on

n = 60

qubits,

|\mathrm{GHZ}_{60}\rangle = (|0\rangle^{\otimes60} + |1\rangle^{\otimes60})/\sqrt2

, is prepared by a Hadamard on qubit

1

followed by

59

CNOTs, each with qubit

1

as control and qubit

j

(

j = 2,\ldots,60

) as target, acting on the initial state

|0\rangle^{\otimes60}

. Someone argues: "This state has one ebit of entanglement across every bipartition that splits the

60

qubits into two nonempty groups, and its state vector has

2^{60}

complex amplitudes. Simulating a state with that much entanglement must, in general, mean tracking exponentially many amplitudes, so no classical computer can predict the statistics of measuring all

60

qubits, whether in the

Z

basis or in the

X

basis, without a computation that scales as

2^{60}

." ((a)) Is the circuit described (the Hadamard and the

59

CNOTs), together with a final measurement of every qubit in the

Z

basis or in the

X

basis, a circuit the Gottesman-Knill theorem applies to? Justify the answer from the gates and measurements used. ((b)) Using the

2n\times(2n+1)

-bit tableau, give the number of bits of classical memory needed to simulate this circuit for

n = 60

qubits, and compare it with

2^{60}

. ((c)) Predict the actual probability distribution over outcomes of measuring all

60

qubits in the

Z

basis: which bit strings occur, and with what probability each? ((d)) Predict the probability of any one, fixed, allowed outcome string when all

60

qubits are measured in the

X

basis instead (an outcome is

+1

-1

on each qubit). ((e)) State, in one or two sentences, what is wrong with the argument quoted above.

Browse the problem library