QTAAS Quantum Mechanics
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Quantum mechanics
Every operation in the quantum coreworld is serialized so that the world's state is modified in only a handful of monomers per step. From the perspective of programs these steps are stochastic and can seem, effectively, simultaneous. In each compartment the QVM records the step (an integer) in which a quantum operation occurs, which operation it is and the qubit(s) involved. The QVM performs whatever simplifications it can and, when necessary, feeds the resulting quantum circuits to a simulator if the QVM requests a measurement. The result of the measurement is registered in the classical state of the QVM and recorded in the qubit's history. The measured states are never too large because—for simplicity—the only permitted quantum operations belong to the stabilizer formalism (with some minor variants). These states are still highly non-local, so, a selective advantage might still be obtained from them. Please see the discussion and the background section on quantum games. During fluctuations overwritten qubits are measured (and swapped), as are qubits bordering between compartments.
Operations under the control of programs
The quantum coreworld permits programs to build very complicated, non-local quantum states. These states are stored as histories of quantum operations and interact with the “classical� biosphere through measurement. Millions of qubits can be present in the same compartment. Practical considerations, however, require that states be limited to only a few thousand qubits; this may not mean that programs run into any limitations of simulation. Programs moved by Brownian motion will lose track of all but their own qubits (the qubits that are part of the program that was moved.) The rest of the qubits in the environment will be measured, so, the maximum expected density of qubits will be related to the density of programs in the coreworld.
Quantum Control Not (QCN)
The control not operation is the only two qubit operation under the control of programs. The simulator stores these operations in the history of both qubits with the time stamp (step) of the operation. When measurement occurs only the subset of qubits which were touched by a control not will need to be considered by the circuit simulator.
Quantum OPeration (QOP)
All single qubit operations are stored in the history of the qubit.
Measurement
When a measurement occurs it might be the case that it can be readily obtained (because the qubit is in a basis state) or it will require the evaluation of a quantum circuit. This circuit is built by considering all qubits that have been in contact with this one (via control not) and ordering all of these by time stamp. If classical operations have flipped the qubit these can be applied by performing the equivalent sequence of operations from the stabilizer formalism. Recallwhere X, by definition, is a classical bit flip. The circuit can be discarded once it has been evaluated and re-computed from scratch from the history. Some qubits—with seemingly complicated histories—may in fact be in basis states. Such measurements can be discarded from the history (since the measurement does not change the state.)
Operations under the control of the environment
At the end of each fluctuation the environment measures qubits in bricks swapped into other compartments; also it measures qubits in polymers reverted to nutrient. These measurements are called decoherence.
Swap
Brownian motion moves qubits at the end of a fluctuation. To do this qubits that have “moved� can be simply swapped by relabeling all references to each qubit with a reference to the other and vice-versa.
Decoherance
Decoherence in the coreworld is implemented as a measurement. It is given a different name simply because it occurs under the control of the environment rather than under program control. Eventually every qubit in a carefully constructed quantum state might decohere. If this happens then the histories of all of these qubits can be purged. Since error correction is possible—the stabilizer formalism is particularly well suited to this—large enough populations of entangled programs may never lose their histories. The accumulation of operations, in histories, that no longer have any casual effect on the state of the coreworld will be a problem that needs further investigation.
Evaluating stabilizer circuits
A stabilizer circuit—consisting of Control Not, Hadamard, Phase and Measurement operations, in any order—can be efficiently evaluated by the QVM. The C code appears in Appendix II (Aaronson and Gottesman 2004). Assume a circuit has been prepared by the QVM based on the operation of programs and the environment they inhabit. No evaluation is required until a measurement or decoherance occurs. Circuits can include gates outside the stabilizer formalism but then require exponential resources in the number of extra gates. Circuits within the formalism are simpler to simulate/analyze and are the only ones considered in this thesis.
