Ph.D. Thesis - Physics

3.3 The bounds on precision
In this section we consider two possible limitations on the precision of the final result. The
first comes from the sampling rate in time used in Step 3 above, and the second from control
errors. We define ε to be the average error in the final result: ε = |∆−∆exp|/∆, where ∆exp
is the experimentally determined value. The number of digits of precision in the final answer
pε scales broadly as log(1/ε). As discussed in Ch. 1, this implies that ≈ 1/ε operations are
required to obtain this precision.
The WBL algorithm requires simulating HBCS for times tq = {0, tq, ...,tQ − t0, tQ},
where Q is the total number of steps, and t0 the step size. The DFT yields an error of
εFT = 2πEmax/Q, where Emax = /t0 is the largest detectable energy. In general, the
length of the simulation is proportional to Q; therefore, the number of gates scales inversely
with the error. A similar problem in Shor’s algorithm overcomes this problem by a clever
way of performing the necessary modular exponentiation [Sho94].
The use of the Trotter formula, as noted above, is also a source of error. In the expansion
above, the error is O(t 3 0 /k2 ). But suppose a first-order Trotter expansion would suffice (as it
may for some other quantum simulation). In this general case, when one wishes to simulate
the Hamiltonian H = HA + HB with [HA, HB] = 0, the expansion to lowest order is
exp (−it(HA + HB)/) = (exp(−itHA/(k)) × exp(−itHB/(k))) k + δ , (3.4)
where the error δ = O(t 2 /k) if [HA, HB] 2 ×t 2 / 2 ≪ 1. The higher-order expansions have
an error δ = O(t m+1 /(k m m+1 ). However, each higher order carries an additional cost of
O(2 m ) more gates. Even though the increase in the number of gates is exponential, the
total time required does not scale so badly. Since the time to implement each gate U(t/k)
is 1/k the time required to implement U(t), the use of the Trotter formula to implement U
requires only 2t total time, regardless of k. It would seem that “Trotterization” does not
increase the inefficiency of the simulation with respect to the precision.
However, we note that future larger-scale quantum simulations will almost certainly
require error correction. The fault-tolerant implementation of U(t/k) actually takes about
the same time as U(t) [Pre98, Got97], whether one uses methods based on teleportation
[GC99] or the Solovay-Kitaev theorem [KSV02]. Consequently, fault-tolerant simulations
using the DFT and the Trotter formula require a number of gates that scales as 1/ε r , where
r ≥ 2.
Although the above argument urges caution for future quantum simulations, we will
restrict ourselves in the rest of this section to the bounds on precision without error correc-
tion. The authors of Ref. [WBL02] predict that the number of gates scales as 3n 4 ∆/ε, where
n is the number of qubits. Can we arrive at an intuition for why this should be the case?
The total number of gates is proportional to the number of gates needed to simulate UBCS,
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