ROM-based computations: quantum versus classical
B.C. Travaglione, M.A.Nielsen, H.M. Wiseman, and A. Ambainis
Quantum research Focused mainly on time requirements
e.g. Shor’s algorithm Deutsch-Jozsa algorithm Grover’s algorithm
Space complexity important also!
Space Complexity Number of (qu)bits required to solve a
problem ROM-Based
Read only memory Writable memory
Space complexity - function of writable memory
This paper Compares space complexity of “error-
free, reversible quantum and classical computation”
Summary ROM-Based computing Universality
1 qubit Quantum ROM computer 2 bit classical ROM computers
Time efficiency
ROM-Based Computation
u1 to uj doesn’t change at any time during computation(Read only bits)
fi is a boolean mapping from a combination of ‘u’s
Classical ROM Computations Sequence of arbitrary classical
reversible gates Gates can be controlled by one of the j
ROM bits FANOUT increases complexity, and bit
reducing activities (e.g. AND) can be simulated.
Quantum ROM computations Similarly, quantum gates controlled by
up to one ROM bit applied to n qubits Note that fi is boolean, so qubits must
end up in computational basis state.
Why only one control? Arbitrary number of controls on
quantum and classical gates can be broken down to gates with 2 and 3 controls respectively
This adds unnecessary complexity as conditional bits must be writable
This constraint does not affect these results
Notation
Notation
Summary ROM-Based computing Universality
1 qubit Quantum ROM computer 2 bit classical ROM computers
Time efficiency
Universality There are 2^(2j) possible distinct
boolean propositions. A universal computer can achieve any
of these. We will show that one writable qubit is
sufficient for quantum case, 2 bits necessary for classical.
Method Well known that AND and NOT are
sufficient to express any boolean proposition.
AND and XOR are also sufficient since “NOT a” can be replaced with “a XOR 1”
Show that any writable (qu)bit can be transformed from |0> to any of the 2^(2j) different boolean propositions
Method Sufficient to show that we can transform
|f> to |f u1u2u3…um > where f is an arbitrary boolean function and
m {1,2…j}
Summary ROM-Based computing Universality
1 qubit Quantum ROM computer 2 bit classical ROM computers
Time efficiency
One Writable Qubit Universal We will use only Pauli matrices:
As well as X-1/2 , X1/2 , Z-1/2 and Z1/2.
One Writable Qubit Universal denotes that the operator W is
controlled by the ROM bit ui.
performs a bit flip iff ui = uj = 1
|f> to |f uiuj >
One Writable Qubit Universal Easy to see with circuit diagram
One Writable Qubit Universal Also Bloch sphere helps visualize
One Writable Qubit Universal We can add more bits to the conjunction
by recursively substituting gates For example, substituting with
Which makes Z essentially controlled by uj and uk, causes our qubit to be flipped iff ui = uj = uk = 1.
One Writable Qubit Universal After substitution we have:
|f> to |f uiujuk > Continuing like this, we can create a
sequence of gates that transforms
|f> to |f u1u2u3…um >
Summary ROM-Based computing Universality
1 qubit Quantum ROM computer 2 bit classical ROM computers
Time efficiency
One Writable Bit Universal? The only operations on one classical bit
are NOT and CNOT. Cannot achieve |f> to |f u1u2u3…um >
with any combination of NOT or CNOT gates with one input. NOT UNIVERSAL.
Are 2 bits universal?
Two Writable Bits Universal We use these 4 gates in our proof
Two Writable Bits Universal These correspond to these equations
Two Writable Bits Universal We will now prove that using these
functions we can transform the inputs |a>|b> to |a>|b u1u2 … uj >
Two Writable Bits Universal
Let S0 denote the N(1) gate above We can show that the sequence
performs the transform
Two Writable Bits Universal If we iterate this to m-1, we will get
Using this , we can come up with a sequence of gates
Two Writable Bits Universal Which results in the transform
This shows that two writable bits is universal by our definition.
Summary ROM-Based computing Universality
1 qubit Quantum ROM computer 2 bit classical ROM computers
Time efficiency
Quantum Time Efficiency Recall that
Quantum Time Efficiency Substituting
we are able to transform
Quantum Time Efficiency This generalizes to replacing each
with
And each with
Quantum Time Efficiency Thus we can take the AND of up to 2k
ROM bits using exactly 4k ROM calls.
Classical Time Efficiency 3 writable bit classical computers can
do this efficiently, but not 2 writable bits. Conjecture: It requires O(2j) ROM calls
for a 2 writable bit computer to perform
Classical Time Efficiency In a non-reversible setting however, the
classical two bit computer requires only j ROM calls as is shown in the following circuit. (O indicates re-initialization)
Conclusion In error-free non-reversible ROM computing: Quantum computers more space efficient
than their classical counterparts only requiring 1 writable qubit to be universal.
Conjecture: Minimal QC can calculate certain boolean functions exponentially faster than the minimal classical ROM computer.