Ph.D. Thesis - Physics

which scales as n 4 , which is proved in their paper. This is multiplied both by the order k
in the Trotter approximation used, as well as total number of steps Q. The fractional error
due to the DFT, as stated above, is ε ∝ 1/(Qt0). We require then that the error due to the
Trotter formula is small compared to this, and may choose k/t0 = 0.1∆. Upon substitution,
we see that εFT ∝ ∆/(kQ), and this proves the WBL bound (up to constant factors).
Arriving at a practical bound on the precision requires taking into account decoherence
processes, because the number of gates that can be performed without error correction Ng
is given by the ratio of the gate time tg to the fastest decoherence time τ. This time,
in practice, is equal to T ∗ 2 (Sec. 2.1). WBL assume a best-case scenario of Ng = 10 5 .
Therefore calculating ∆ to a precision of ∆ = εFT is possible for a maximum number of
n = 3 10 5 /3 ≈ 10 qubits. However, given the specific scalar couplings (tg ≥ 60 ms) and
decoherence rates of our system (T2 ≈ 10 s), we should be able to saturate this bound for
n ≤ 4 qubits.
Of course, the above considerations assume perfect control pulses, which in reality do
not exist. We now take a look at how control errors can affect the precision. In the static
field of the NMR magnet, B0ˆz, and in the absence of rf pulses, the unitary evolution is
given by
⎛
UZZ = exp⎝−i
π
2 JijZiZjt
⎞
⎠ . (3.5)
ij
Normally, we make the approximation that the time required to rotate a spin by π
radians is much smaller than the delay times during which no rf is applied, and the system
evolves freely according to Eq. 3.5, that is, tπ ≪ td ≈ 1/Jij. This is justified since tπ is
O(ms), and 1/Jij is O(10 ms). When this approximation holds, the rf pulses are treated
as δ-functions in time, implying that evolution under the ZZ interaction is assumed not to
occur during this time. As tπ approaches td, these control errors can be mitigated to some
extent, but as the number of pulses becomes greater and greater, this effect also becomes
more pronounced.
In the experiment, we will explore whether we can saturate the theoretical bounds on
the precision, even given faulty controls. First, though, we discuss the actual hardware that
was used in our experiment.
3.4 The NMR system
We now turn to a description of the experimental system used to implement the WBL
algorithm. This section contains a description of all the key components of the NMR
system used in our work. Here we describe the hardware that implements all the operations
discussed in Sec. 2.1.
The NMR system consists of an 11.7 T magnetic field, oriented vertically, which provides
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