How to Build Your Own Quantum Computer

Lecture 19: How to Build Your Own Quantum Computer Department of Mathematics, MIT 1 The DiVicenzo Criteria The DiVicenzo Criteria list four things required for quantum computing: robust qubits, a universal gate set, a fiducial input state, and projective measurements. 1.1 Robust Qubits A quantum qubit is based on 2 level quantum systems. One must also remember that a tensor product Hilbert space is needed; for example a harmonic oscillator is not a tensor product space and therefore makes a bad quantum computer. In general, a two level atom would make a good quantum computer. One must also have a long coherence time. This is characterizes how the environment interacts with your ideal qubit system. The imperfect qubit system will have many states besides the two you are interested in. One can think of decoherence as the effects of all the interactions outside your idea set of interactions. |Ψ(t)� = e−iHt |Ψ(0)� ⎡ a b g ... ⎢c d h ⎢ H = ⎢e f i ⎣ .. .. . . ⎤ ⎥ ⎥ ⎥ ⎦ In the Hamiltonian above, the elements in bold represent the your ideal set of interactions, and everything else is the non­ideal part that causes decoherence. There are many sources of decoherence. Gravity causes decoherence if one states weights more thant the other. There may be stray long range fields, typically associated with charge. There can be leakage into larger Hilbert spaces; a two state atom may have higher energy levels that the state can move to. However, there does exists true finite spaces in nature, such as spin. There are two measurements of decoherence, T1 and T2 . T1 is called the “longitudinal coherence time,” or the “spin lattice time,” or the “spontaneous emission time,” or the“amplitude damping.” It measures the loss of energy from the system. One can do an experiment to determine T1 . First 1 2 T1 T2 1 probability of being in |0> state probability of being in |1> state 1 1/2 time time Figure 1: expected results for experiments for T1 and T2 initialize the qubit to the ground state |0�. Then apply X = |0� �1| + |1� �0|, and wait for time t and measure the probability of being in the |1� state. We expect an exponential decay e−t/Ti . T2 is called the “transverse coherence time,” or the “spin­spin relaxation time,” or the “phase coherence time,” or the “elastic scattering time,” or the “phase damping.” One can do an experi­ ment to determine T2 . First initialize the qubit to the ground state |0�. Then apply the Hadamard |1� √ , wait for time t, apply H again, and measure the probability transform H to get the state to |0�+ 2 of being in the |0� state. We expect that the measurement goes to 1/2 after a long time because after a long time, most likely something popped the state into either |0� or |1� state, which after √ the H transform sends the state to |0�±|1� . 2 In general T1 > T2 . 1.2 Universal Gate Sets There are many different universal gate sets. We have seen that CNOT and single qubit gates form a universal gate set. CNOT, the Hadamard gate, and π/8 gate is another universal gate set. In practice experimentalists use controlled Hamiltonians that they turn on and off for certain time intervals. H = Hsys + P1 (t)X + P2 (t)Y... For example, for an NMR setup, we may have π� Jσz ⊗ σz + P1 (t)I ⊗ σx + P2 (t)I ⊗ σy ... 2 In the real world, there is no such thing as time dependent Hamiltonians. So how do we perform a sequence of operations? It is a just an approximation; fundemantally we will always have decoherence and will need fault tolerance. What happens is that our classical controls are actually quantum systems, and we must take into action the back action of the control system on our system. P1 (t)X is just an approximation; in reality, we have a Jaynes­Cumming type interaction Hamiltonian: H= H = �ωN + δZ + g(a† σ− + aσ+ ) where σ� = X±Y 2 . In less abstract terms, there is decoherence that results after a photon interacts with a qubit because the photon will carry away information about the state of the qubit. 3 2 Implementations In general, the challenge of quantum computing lies in the fact that quantum systems have short lifetimes, and that we need to control it externally. System NMR Ion Trap Dots Microwave Cavity Optical Cavity T (sec) 102 to 108 10−3 10−6 100 10−5 Table 1: Some relaxation times for different implementations 2.1 Cavity QED The Hamiltonian is described by atom, photon, and atom­photon interactions. The qubit is the sin­ gle photon. Gates for π/8 and Hadamard are implemented by beam splitter and the like. Turchette managed to achieve a control Z gate. 2.2 Ion Trap The Hamiltonian is described by spin, photon­phonon (vibrational mode ) atom­photon­phonon interactions. The qubit is the atom (spin) and the phonon. The Deutsch­Joza algorithm has been implemented with ion traps. The system has a lifetime of O(1 − 100) (order of) milliseconds. Pulsed lasers are the universal gate set. The challenge is to get a stable O(10) Hz laser and cooling the ions to absolute zero. 2.3 NMR The Hamiltonian is described by spins, spin­spin, and external control photon interactions. The spins interact with chemical bonds. The gates are implemented by radio frequency magnetic field pulses. Factoring with 6 qubits has been accomplished. In terms of robustness, 1 H, 13 C, 9 F , 14 N has coherence times of O(1) seconds. 2.4 All Silicon Quantum Computer This implementation implants individual atoms into silicon surrounding and controls the atom electronically. It has a coherence of 100 ms at T = 9.2K. This implementation is scalable and uses current techniques of classical computer engineering. 3 Quantum Cryptography Currently two companies make quantum cryptography systems. The purpose of quantum cryptog­ raphy is to make it more likely to detect an eavesdropper. They are based on the fact that an eavesdropper measuring a quantum system transmitted will collapse the system. 4 We can relate various techniques of quantum computing to features of it’s implementation. Data compression is related to cooling. Error correction is related to control and T1 and T2 . Noisy coding is related to precision of measurements. Cryptography is related to entanglement and non­locality.