Meetings
Meeting 27/09/16
Discussed the LCN
Connection of Ancilla Asisted to State Injection?
 Canonical state injection is a Clifford isometry + pauli measurement + controlled correction
 For what states rho can this map be unitary?
 What rotations did the ancilla assisted method allow?
Is there a limit to ‘deterministic’ injection to equitorial states only?
 e.g B&Ks probabilistic conversion step for
General theory here, what is the limits of these Clifford isometries?
 Recall isometry = generalisation of a unitary
 Unitarity restriction is strong
 Analytic answer here?
 Measuring a bunch of Paulis => PauliBased computing
 Projection on to a coded version of
 Face state method?
For what states can we do this deterministically? How does U depend on the copies of the input state? Again, question of deterministic nature
Start with a two qubit example, see if we can see for what states this is a unitary map….
 Only interested in states identical up to clifford equivalence, so where are they on a quadrant of the Bloch sphere; 1/3 of a face (consider the gate)
 Single qubit isometry looks like projection on to a Bell state?
 Try and understand it exhaustively for the two qubit case
 Looks like projection on to a twoqubit code and then a decoding…
 Look at decoding circuits yeah? Remember what they look like
 Only a 00, 11 code or a 01 10 code… All others are clifford equivalent
 Check equivalence to this Bell circuit up to CLiffords before the isometry or after the correction!
 All things which act inside the isometry are nontrivial maybe?
 We expect this to work for diagonal only
 WHat happens beyond two qubits?
 We don’t want to assume determinism
 Focus on the face state numerically and analytically
 Treat B & K as a hint
Meeting 01/11/16
Met with Dan to discuss current progress on generaising state injeciton
Semi Clifford Injection
 Chuang’s results about how any SC gate is injectible and how much of the resource space they cover
 Seems likely to be the best of the best given the appearance of the Cifford hierarchy in the CORRECTION that makes the circuit deterministic
 Two avenues of attack: Diagonalisation OR remvoing the correction and using a procedure to take the nondeterministic outcome determinstic
Nondeterministic Correction
Based on a concept developed by Nielsen in 99 (Phys Lett. A, Quantum Computing with Measurements and Quantum Memory), which is itself inspired by the notion of a random walk on a circle. We can use the fact that the ‘failed’ injection realises an operation that looks like to use this formalism. For diagonal operators such that , this converges in steps which is p. good? (Compare to teleportation depth)

Matrix Product States
Related to a paper on making nondeterminstic gates deterministic in measurement based QC, which uses the MPS formalism. We could maybe employ this here though the entangling operation will be HIGHLY nonclifford so the resource advantage is quesitonable

Port Based Telportation
Paper from Jonathan and a Horodecki, uses multiple EPR pairs and a CSWAP to realise teleportation AND we can reclaim some entanglement afterwards!
We could look to extend that by first using it to teleport ‘through’ unitaries, and then using the same generalisation of Chuang et al. to use it for portbased injection.
NOTE that this could be considered similar to Fowler’s Time Optimal Quantum Computing method where he allows for the destination of the teleportation to be selected based on stabilizer measurements.
Diagonalisation Problem
Based on the procedure discussed by Bravyi & Kitaev in their original paper on magic state distillation. They introduce a probabilistic procedure for diagonalising the T state based on even parity measurements, such that the phase on the 11 component is i, and this can be converted to a diagonal state using a Hadamard operation.
The requirement seems to look like taking two copies of a state and doing a Hadamardlike rotation .
Also read through their argument on using these diagonal ancillae. They are not using the stateinjection formalism AT ALL and instead seem to be realising controlled rotations using the random walk method of Nielsen we discussed above.
Meeting 13/12/16
Met with Dan to present my ideas on ‘equatorializing’ states in a way that makes them consumeable in a Repat Until Success injection scheme
Equatorialization Via 221 encoding w/ H’ gates

This idea was not especially successful. The repeat tensor and parity projection has the effect of pushing the amplitudes of the system away from the ‘unstable’ fixed point of .

Fideility of this process was generally poor which is testament to the weird success condition for the phases cancelling out.
Equatorial States
We focus instead on the idea of trying to create states on any of the three main equators, the XY, XZ or YZ planes. These are all Clifford equivalent, and give us three classes of injectable phase states
We will focus singularly on the real XZ plane (class 3) for simplicity as these are all Clifford equivalent.
Conjugate States
Consider tensoring two copies of a state , and selecting only the even parity states. We end up with the reduced state (after disentangling)
If we instead take a product of the state with its conjugate, then the phase cancels and we are left with an equatorial state with an angle given by .
How realistic is it to have access to these conjugated states? We are operating in the regime of a resource theory, where we assumed access to some preparation scheme for producing (mostly) pure states. If we are focusing on rotations, we can consider inverting the Y rotation. If we instead have a distillation scheme, we can instead invert the Y stabilizer eigenvalues. Thus, it’s reasonable to assume access to pure states.
We can alter the angle or the operaiton we realise by acting additional Clifford operations on our resource states.
Twirling
An alternative option for equatorializing states would be to ‘Twirl’ them, which is to randomly apply some subgroup of Unitary operators such that the density matrices you get out demonstrate some of the strucutre of that chosen subgroup.
For example, states in the XZ plane are stabilized by the Hadamard twirling, which will map any mixed state in to this plane. The question is then how we can arrive at a pure state we can use after doing the Twirling?
Stabilizer Rank as a resource measure
Now we have this picture of using a pair of arbitrary states as a resource, we can start to ask about how the Stabilizer rank compares for them. In particular, Stabilizer Rank is monotonically decreasing under Clifford operations and Pauli measurement, making it interesting as a possible resource measure for quantum advantage. What’s more, the production of the equator state is an irreversible process. So it would be interesting to study how () behaves in this process.
Stabilizer Rank Chat 14/12/16
Chat with Earl about his updated version of the stabilizer rank bounds, extending the Bravyi Gosset method to all C3 resource states.
Note on Stabilizer Ranks:
Earl’s method amounts to generating stabilizer states from random codes. This method is slightly odd as it is discontinuous. The number of random codes needed to find an approximate stabilizer rank decomposition scales as 1/f, where f is defined as the maximum fidelity between the target state and a stabilizer state, weighted by the approximation accuracy
This second term goes to infinity as we make our approximation exact. This suggests the random code technique only applies for approximate decompositions?
What is perhaps more interesting for stabilizer rank as a resource measure is looking at the problem of finding exact decompositions. In this note, Earl suggests a method based on subgroups of the Clifford group and defines the related notion of a ‘Clifford’ magic state, a naturally C3 state stabilized by Clifford group operations. The stabilizer rank of exact decompositions is then bounded by the quotient of the order of the two groups. This induces some curious effects not previously outlined, including descriptions of unitaries that need seemingly unbounded numbers of T gates to synthesise?
Does the structure of the Clifford group involved hint at a way to DISTILL these states?
Ideas on how to progress

Symmetry?
The arguments used to find decompositions for H and maybe for F states are based on a symmetry of the pair of states, e.g. for the H state we have a state that is symmetric under a unitary and its transpose/conjugate; it is unclear precisely which given the nature of the Hadamard operator… Symmetry of states under where is some linear map is related to the definition of the state by the ChoiJamiolkowsky isomorphism. Pursuing this seems to maybe have phase issues as a unitary and it's transpose is not necessarily unitary…? We could look at this for face states though as Dan wanted back in the MRes days. 2. Expanding the group:
The trick Earl uses for exact decompositions will always give an order that is a power of two due to the nature of abelian Clifford subgroups. There are a few options here. One could be to look for symmetry in the larger stabilizing Clifford group. The other could be try and look for nonabelian extensions of the groups by including e.g SWAP operators. Initially that would seem we could increase the group order by at least for qubits.
(I have a note here that says sum over orbit w/ Reflection Symmetry?)
Alternatively, we could think of the fact that these stabilizer states are always even superpositions and see if that symmetry can help us to identify the decompositions…
Immediate focus is to look at formalising the BSS decompositions, which focus on random walks over realvalued stabilizer states. They can be understood as cancelling amplitudes (see stabilizer rank notes) over different sets of strings with equal amplitudes.
Meeting 23/1/17
Notes on methods tried since December
We can find explicitly the decompositions used by Bravyi, Smolin and Smith . In particular, the write the state in a weird normalisation
Where the amplitude is given by the Hamming weight of the computational basis string. These amplitudes are all real, so they then focus on real valued stabilizer states built out of sets of strings group by Hamming weight and by graph states.
There is no apparent structure to these decompositions that could easily be algorithmized... Dan suggested a scheme based on trying to glob pairs of states together to 'known' computational stabilizer states, e.g how a combination of 00,01,10,11 is equivalent to two plus staes, but it's unclear how useful this approach would be.
BSS use random walk methods, which seem perhaps the most efficient. Other methods grounded in finding sparse decompositions in the 'stabilizer basis' require us to generate the exponentially many stabilizer states first, and then do either brute force work or rely on 'greedy' estimators like Smoothed L0.
It would be desirable, then, to have fast, well optimised code for generating stabilizer states and then plugging them in to these 'solver' methods.
Product States and Maximising Stabilizer Rank
In general, the size of the Stabilizer rank (for magic states in particular) is controlled by two things: the symmetry under Clifford group operations, and the fact that our state we consider is generally a tensor product of n copies. There are a few observations to consider

The Stabilizer Rank is monotonically decreasing under 'Stabilizer' mechanics. This suggest that any state we can prepare (exactly) using circuit synthesis built out of n injected Tgates cannot have a larger stabilizer rank than the magic states.
Relatedly, we can envisage a connection between the behaviour of approximate stabilizer rank decompositions (as used by Bravyi and Gosset and discussed by Earl), and the approximate generation of an arbitrary state using Clifford+T synthesis.

How might we maximise the Stabilizer Rank?
It seems intuitive that this would be by generating random states, and this is indeed suggested by my work with random states in the MRes. It would be interesting to compare how this behaves then for Haar random states, either taking n copies of one state, or n unique random states, or a random nqubit entangled state.
Other observations from the meeting include a few points about what we can learn from the structure of the stabilizer rank. For example, consider the fact that two copies of the T state have the same stabilizer rank as one, but that the classes of state we can generate using Clifford+T synthesis is distinct for 1 and two T gates. This suggests that all states generated by 1 & 2 T gates have some common property that is 'promoted' when we add a third T gate, and this is worth exploring algebraically.
We would also like to explore a connection between a unitary U used to generate a state from an arbitrary stabilizer state, and the location of that Unitary in the Clifford hierarchy. Relatedly, how does Clifford+T synthesis let us 'walk' through the Clifford hierarchy?
Meeting 24/02/17
Since the last meeting, I have reworked the code I developed under my MRes project to the new package stabilizer_search, and this package is now amture such that the simualted anneal method can be used to find decompositions on >2 qubits. I also explored some analytical ideas related to the tensor product of arbitrary states, that are discussed in the notes on the Stabilizer Rank project.
What states have 'known' stabilizer rank?
Consider explicitly constructing combinations of two stabilizer states with coefficients . We can use the requirement that the resulting state is a product state, and thus the reduced density matrix of each qubit is pure. This should give a quartic to solve in terms of the coefficients. This would allow us to understand what properties of product states give a small stabilizer rank, and similar methods could be used on more qubits to examine arbitrary states.
Related to the above question of the connenction between the Seilinger synthesis method and the stabilizer rank, it'd be interesting to construct states with an explicit stabilizer rank and to see what their coverage in the space of states is.
Hypergraph staes and stabilizer rank
Hypergraph states are constructed like grpah states, but using 'hyper edges' between multiple vertices such that the hyper edges define multicontrolled Z gates. In particular, if the hyperedges are buiult out of 3 qubits, they are constructed by the CCZ gate whcih lives in , and is thus nonClifford. It would be interesting ton determine the stabilizer rank of copies of CCZ, much likes copies of roots of T, and see how it grows.
Symmetry acting on stabilizer states
We have to consider the consequence of stabilizer rank being invariant under certain symmetries. This is an idea that I briefly explored in the context of tensor products of arbitrary states, where we know that the decompositions must be invariant under local single qubit operations , and the group generated by these, and under permutations of qubits.
Relatedly, for the case of states, the unitary symmetry operation will be a Clifford operation, and thus the amplitudes in the decomposition remain invariant but in general we get new stabilizer states out. The space spanne by the states under the orbit of the Clifford symmetry group must always contain our target state (this is generally true but easier for Clifford symmetries as we never leave the set of stabilizer states and the decomposition amplitudes are constant). Can we explore this to extend the 'quotient group' idea that Earl sets out in his note?
For nonclifford operations, the the action of the untiary on the stabilizer states is pretty much always going to produce something weird. The one interesting property we have here is that these states will also admit a stabilizer rank decomposition, and there must be a way to combine the states in these decompositions to get out a new expression with the stabilizer rank left invaraint. Stabilizer states combine in ways that are not entirely obvious, so it might be worth reading about their 'stabilizer' operations (the mutually unbiased operator basis).