Last summer I started thinking about my master’s thesis on weak measurements. I’ve been keeping an eye out for interesting weak measurement papers for a while and have had the opportunity to referee a few papers on the subject that forced me to keep up to date. I started playing around with some weak measurement ideas when Raymond Laflamme (one of my current supervisors) suggested I give a short introduction at the next group meeting. The biggest question at the end of this short introduction was “can we do this in (liquid state) NMR?”. My first response was an outright `NO’, because any interesting weak measurement experiment would require post-selection (see below), a very difficult task in an ensemble system like NMR. After some serious thought I realized that the solution was actually very simple. What I found amazing was that the experimentalists were able to perform the experiment immediately, in-fact these guys can perform any small quantum circuit without too much trouble. The result was the first weak measurement experiment that did not involve any optics. The paper was published in NJP (open access) and a video-abstract is available on the NJP website and youtube.

This month I also taught a short four lecture module on weak measurements and the two state vector formalism as part of QIC 890. But I will keep the discussion of weak measurements to another post. For now I will explain the trick used in the NMR experiment. That will require me to first explain some issues regarding ensemble quantum computers.

## Ensemble quantum computing

Today we don’t know what a quantum computer will look like. We don’t know what it will be made of and we don’t know how it will work. While from a computer science perspective all architectures are the same, that is they can solve the same problems, from a practical perspective they are quite different. Nevertheless in most cases we like to think of an abstract quantum processor in a similar way to a standard processor, in terms of circuits.

The circuit accepts a classical input, a series of zeros and ones, encoded in quantum bits. The circuit itself is a sequence of operations on those quantum bits. These operations are reversible (unitary) but otherwise they can be quite general. At the end, some of the quantum bits are measured in a specific way and a classical output (a series of zeros and ones) is produced. This output is usually not deterministic so the program can produce different outputs for the same input. Although this seems like a flaw it is not, as long as the probability for an unwanted result is low.

In liquid state NMR the quantum bits are the nuclear spin degree of freedom of single atoms on a molecule. The molecule is the processor and the natural electromagnetic interactions inside the molecules are supplemented with controlled external fields to produce the dynamics (i.e the gates). Control in this system is very good but there are a number of downsides. The main downside is that the signal is very noisy. To overcome the issue of noise a large number of molecules are used. This means that a large number of identical processors are running in parallel.

One of the drawbacks of running the computation on an ensemble of identical processors is in the readout stage. The final measurement is an ensemble measurement and the result is a statistical average. Why is this bad? Let us say for example that we are running a classical computation on two bits with two possible results. Half the time the result is 0,0 and half the time it is 1,1. Now if we read the average on each bit we get that each bit is 1 half the time and 0 half the time so on average it is 1/2. But this average result 1/2,1/2 is also consistent with an output which is 0,1 half the time and 1,0 half the time. So we can’t distinguish between these results.

Liquid state NMR is not the only system where this kind of ensemble paradigm applies and it is quite possible that ensemble quantum processors will be the way to go for quantum computing, at least in the short term. Liquid state NMR is also the current record system, with good control of 12 qubits. It is therefore not a surprise that people have come up with methods for circumventing the shortcomings of ensemble readouts. Going back to the example above it is possible to have a third bit register set to 1 if the first two are equal and 0 if they are not. This will distinguish between the first and second scenario above. In the first case we will have 0,0,1 half the time and 1,1,1 the other half while in the second case we will have 1,0,0 half the time and 0,1,0 the other half.

## Post selection and weak measurements.

In the case of post selection we want to read the average result of the first (quantum) bit but only in the case where the second one is in a specific state (say 0). So if we have 0,1 one third of the time 0,0 one third and 1,0 one third we should to read out 1/2, the average of the first bit only in the two cases where the second was 0. A similar situation exists when we want to get an interesting result for a weak measurement. The reading on the measuring device must be post-selected according to the state of the measured system.

To perform the post-selection we used a (seemingly) non reversible operation. Sticking to the example of classical bits above our algorithm worked in the following way. We want to post select on the cases where the second bit is 0. To achieve this we perform an operation that randomizes the first bit if the second bit is 1. When we get the averages at the end we know how many times we got a random result (by measuring the second bit) and how many times we got a `real` result. Using this information we can get the statistical average of the post-selected states.

The quantum case is a little more involved but the basic idea is the same. This trick allowed us to perform the weak measurement experiment with post selection and get strange results such as complex values and values in outside the normal range. The method we used for post-selection goes beyond weak measurements. We are currently thinking about other weak measurement experiments as well as other experiments that involve post-selection. The advantage is that we can control bigger systems than anyone else (by we I mean the experimentalists, I can’t control anything).

This was also my first collaboration with experimentalists. I’m looking forward to more.

*Experimental realization of post-selected weak measurements on an NMR quantum processor,*

*Dawei Lu, Aharon Brodutch, Jun Li, Hang Li, Raymond Laflamme, NJP 2014.*