Computing with highly mixed states

Abstract

We consider quantum computing in the one-qubit model where the starting state of a quantum computer consists of k qubits in a pure state and n − k qubits in a maximally mixed state. We ask the following question: is there a general method for simulating an arbitrary m-qubit pure state quantum computation by a quantum computation in the kqubit model? We show that, under certain constraints, this is impossible, unless m = O(k + log n). INTRODUCTIONIdeally, a quantum computation is a sequence of local unitary transformations applied to a register of qubits which are initially in the state |0 n ; followed by a measurement. Initializing the state of the quantum register is the biggest challenge in NMR quantum computing (which is perhaps the most advanced technology in terms of the scale of experiments performed to date [2]). The difficulty is that the register is actually initially in (approximately) the binomial distribution over pure states |x , in which each qubit is independently in the state |0 with probability ; the currently achievable polarization ǫ is quite small. There are currently two ways of implementing quantum computation in this technology. The first is used in current experiments [5; 3], but does not scale beyond several qubits -the output signal decreases exponentially in the number of qubits in the quantum register. The exponential decay in signal to noise ratio in any scheme that embeds virtual pure states on an n-qubit quantum computer with one clean qubit is unavoidable, due to the result [9].liquid NMR [11]. An intriguing third possibility was raised in [7]. Suppose we start with one qubit in the pure state |0 in tensor product with n − 1 qubits in a maximally mixed state (i.e. in a uniform distribution over basis states |x ). Is it possible to simulate general quantum computation by effecting a sequence of elementary quantum operations on this register? If the answer were affirmative, this would yield a procedure that would both scale and be currently feasible using the scalable initialization procedure to convert the initial binomial state to a state where the last n − 1 qubits are maximally mixed and the first bit has high polarization (the strength of the output signal is now proportional to this polarization). This is the question we focus on in this paper. It is easy to see that if all n qubits are in the maximally mixed state then no computation is possible. This is because applying any unitary transformation to this mixture leaves it invariant. This simple argument stands in striking contrast to the difficulty of the seemingly very similar case, in which just a single qubit is in a pure state, while all the others are maximally mixed. Since the initial state of the register is completely specified, the only real input in this model is the sequence of elementary quantum operations. So, given a quantum circuit C which we would like to simulate on an input x, we wish to know whether there is a sequence of elementary quantum operations on the n qubit register, which first prepares a...

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