Ph.D. Thesis - Physics

Digital
Analog
Classical Quantum
Space: 2nO(log 1/ǫ)
Time: T22n Space: 2n Time: T
Precision: O(log 1/ǫfix)
Space: n
Time: Tn 2 O(1/ǫ r )
Space: n
Time: T O(1/ǫ)
Table 3.2: A comparison of the resources required for digital and analog simulation of
quantum systems, using both classical and quantum systems, and taking into account the
precision obtained. We consider a system of n qubits simulated for a total time T. The error
due to projection noise is denoted ǫ, while the fixed error of a classical analog computer is
denoted ǫfix. Here r ≥ 2 when error correction is required.
the quantum simulation is still more efficient than a classical algorithm, provided no
efficient classical approximation exists.
2. In general, use of the Trotter formula or some related technique will be necessary
in digital quantum simulation, as most non-trivial Hamiltonians have mutually non-
commuting terms. We have shown that, when fault tolerance is required, this process
increases the number of required gates by a broadly exponential factor. The conclusion
here is roughly the same as in item 1: this class of simulations is inefficient with respect
to the precision, but moreso when error correction is needed. This result is included
in Table 3.2.
3. Control errors are a problem that afflicts any quantum simulator. The type that
appear in NMR arise from the “always-on” scalar coupling interaction. Although
for small systems it is relatively easy to compensate (to first order) for these errors,
and thereby bring the solution to within the theoretical bound, it will not be so
straightforward in larger systems. Fortunately, in scalable systems such as ion traps,
the two-body interactions are controlled more directly by the experimenter. There
will be control errors in that system as well, but of a different sort.
4. All quantum simulations are limited by natural decoherence times. The bound on
the number of qubits that may be used to obtain an answer to within a given error
depends on the number of gates that may be performed within the coherence time,
unless error correction is used. Error correction becomes necessary for larger systems,
but then the overall scaling properties become less favorable, as explained in items 1
and 2.
We would like to point out that the above results apply to digital quantum simulation
generally, regardless of whether the classical discrete Fourier transform (DFT) or the quan-
tum Fourier transform (QFT) is used. Our results apply to both for the reason that they
deal with the implementation of the simulated Hamiltonian; whether the DFT or QFT is
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