A relational quantum computer using only two-qubit total spin measurement and an initial supply of highly mixed single-qubit states

We prove that universal quantum computation is possible using only (i) the physically natural measurement on two qubits which distinguishes the singlet from the triplet subspace and (ii) qubits prepared in almost any three different (potentially highly mixed) states. In some sense this measurement is a ‘more universal’ dynamical element than a universal two-qubit unitary gate, since the latter must be supplemented by measurement. Because of the rotational invariance of the measurement used, our scheme is robust to collective decoherence in a manner very different to previous proposals—in effect it is only ever sensitive to the relational properties of the qubits.


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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

Introduction
There has been considerable effort directed towards understanding how the measurements can be utilized in quantum computation [1,2]-either as part of error correction, or as a method of replacing some (or all) of the coherent processes of the standard unitary circuit model.With respect to this latter program, it is important to try to develop schemes involving measurements that are physically natural.
An example of a physically natural measurement is the parity measurement on generic bosonic systems-a measurement which compares the state of two systems as to whether they are the 'same' or 'different'.Parity measurements can be performed using Clifford operations 3 , and have formed some part of various measurement-based schemes.Recently it was shown that certain non-deterministic parity measurements, along with single qubit unitaries, are universal for quantum computation [3], and the proof allows for a dramatic simplification of the resources required for linear optical quantum computation.
In this paper, we will focus on a different physically natural two-qubit measurement that is not a Clifford operation (we will see why this is the case later).Abstractly, the measurement is composed of the projectors where |ψ − is the singlet state.As a measurement on two spin-1/2 systems, this rotationally invariant 'J-measurement' is one of total angular momentum-a projection onto the singlet or triplet states according to whether the total angular momentum is 0 or 1 respectively.J-measurements are physically natural, primarily because in a wide variety of atomic and solidstate systems the natural interaction Hamiltonians have different energies for the singlet versus the triplet states.
Various results on the universality of J-measurements for quantum computation can be readily obtained.For instance, it can be shown that J-measurements, single-qubit measurements and unitaries (in particular, say, just Hadamard and phase gates), and systems initialized in the computational basis state |0 , are universal.In fact, it can even be shown that any two outcome measurements composed of projectors {|φ φ|, I − |φ φ|}, for an arbitrary two-qubit state |φ , is universal under similar conditions.
However, here we will show the much stronger result that Theorem 1.Quantum computation can be performed using (a) two qubit J-measurements, and (b) any (polynomially large) supply of single-qubit mixed states prepared along three linearly independent Bloch vectors.
Our scheme has a number of interesting features: • We do not require single-qubit measurements, nor do we require the initial supply of states to be pure.In fact, the initial qubit states can be very highly mixed, as long as none of them are maximally mixed.To date, all other schemes either require more than one type of measurement and/or require initially pure states and/or require joint measurements on more than two qubits.
• The J-measurement is the only dynamical object used in the computation.It is therefore 'more universal' than a universal two-qubit unitary operation, which must be supplemented by some form of measurement.• The J-measurement is rotationally invariant-the measurement is sensitive only to the relative state of the systems involved (in fact, it is the optimal measurement for determining relative information of two qubits [4]).As such, the computation is naturally robust to random collective rotations-not because the states are rotationally invariant, as is the standard procedure for protecting against such decoherence, but rather because the dynamics are robust.
We will prove theorem 1 utilizing the cluster-state model of quantum computation [2].We focus on the simplest path to prove universality-for many of our constructions there exist more efficient, but less transparent, procedures.We will heavily use the following three useful properties of the J-measurement, proofs of which will be given later.

Property (i)
Purification.Given a supply of single-qubit mixed state ρ with Bloch vector r 0 r, r 0 0, and the capacity to do J-measurements, it is possible to prepare single-qubit states with Bloch vector (1 − )r using constant resources for any fixed > 0. (The fact that such purification can be achieved is an easy way to see that the J-measurement is not a Clifford operation-it is known that Clifford operations cannot be used to purify arbitrary Bloch vectors, see for example [5,6].)Property (ii) Programmable single-qubit measurements.Given 2 n − 1 copies of a pure singlequbit state |φ , a projective measurement of an arbitrary qubit onto the orthogonal pair of states {|φ , | φ } can be simulated by J-measurements, with fidelity that goes as 1 − 1/2 n .(This measurement does not collapse the qubit being measured in the same way as a standard projective measurement would; however it (asymptotically) collapses a remote system with which the measured qubit is entangled in the standard way, and this suffices for our purposes.)

Property (iii) Creation of maximally entangled states. It is possible to create all four Bell states (|ψ
)), as well as all GHZ states of the form: Using the above properties, we now prove the theorem.
Proof of theorem 1.From properties (i) and (ii) it is clear that the single-qubit states prepared, and the single-qubit measurements implemented are only asymptotically 'sharp'.However, in [7] fault-tolerant procedures for cluster state computation have been demonstrated for certain types of noise, including situations in which the cluster state is prepared non-deterministically.
The types or error incurred by our scheme's inherent imperfections fall into the class of error models considered by [7], with at most minor restrictions on the topology of the cluster states that we are allowed to make.Hence, there exists a finite fault tolerant threshold that we need to obtain.This threshold can be obtained with constant effort, and hence for now we will proceed as if properties (i)-(iii) involve no imperfections whatsoever.Given this assumption, the proof proceeds via a sequence of primitive operations, and so we will first demonstrate how each of these primitives may be achieved.In the rest of the paper, we will frequently omit normalization factors from our equations in order not to clutter the notation.
Creating 'flipped' qubit states.From a qubit-mixed state with Bloch vector r, we can create a 'spin-flipped' qubit with Bloch vector − r/3 using the following procedure (an optimal Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT cloning/spin flipping): take an ancillary pair of qubits in a singlet state (which is readily obtained), and implement the J-measurement on one member of the singlet and the qubit to be flipped [8].If the J 1 outcome is obtained, which occurs with probability 3/4, it is easily verified that the reduced density matrix of the unmeasured singlet qubit now has Bloch vector − r/3.
Creating arbitrary qubit pure states.By flipping each of our original set of three types of qubits with linearly independent Bloch vectors, we can efficiently create qubits with six different Bloch vectors.Probabilistically mixing these states allows us to generate any mixed state that lies inside the polyhedron that has these six states as its vertices.Simple geometrical considerations show that such a polyhedron necessarily contains a sphere of finite radius centred at the origin.Any such states can then be purified (using property (i)), leading to the creation of arbitrary pure states.
Creating two-qubit cluster states.By property (iii) we can create all four Bell states.We can now create the maximally entangled two-qubit cluster state |0+ + |1− as follows.Take four qubits in the state |ψ − 12 ⊗ |φ + 34 and perform a J-measurement between qubit pairs 1,3 and 2,4.In the event of obtaining the J 1 outcome on both measurements (which occurs with probability 1/2), the four qubits are collapsed to the state |φ − 14 |ψ + 23 − |ψ + 14 |φ − 23 .It is readily verified that now performing a single-qubit measurement on qubit 1 in the |0 , |1 basis, and qubit 4 in the |± basis, collapses qubits 2,3 to a two-qubit cluster state, regardless of the outcome 4 .
Redundant encoding of cluster states.In order to create larger cluster states, we will need to utilize the concept of a redundant encoding of a given qubit in the cluster [3].Such an encoding is one in which extra physical qubits are appended 'in parallel', such that they are still considered to be part of the logical encoding of a single-cluster state qubit.More precisely, a generic cluster state can be written (|X |0 + |X ⊥ |1 ), where we have singled out one qubit from the cluster state.A four-qubit redundant encoding of this qubit would be 4 ).Note that it is not necessary for the states in a redundant encoding to be the 'same' qubit state in parallel-the two redundant strings need only to be orthogonal at each position, and in particular could be any bitwise orthogonal strings of {0,1}.We will soon show how such a redundant encoding may be achieved.At the point of performing a cluster computation, unwanted redundantly encoded qubits can be removed by measurement in the |± basis-for example, measuring and then discarding the last qubit in the redundantly encoded cluster (|X |0 |0 + |X ⊥ |1 |1 ) gives one of |X |0 ± |X ⊥ |1 , which are the same as the unencoded cluster state up to at most an unimportant Z rotation on one qubit.
Fusion of small clusters into larger ones.Consider two independent cluster states, with one-qubit singled out from each, such that the states can be written as A fusion operation produces one of the states or any states obtained from these via a Pauli gate applied to the singled out cluster qubit.It can readily be verified that these two states are essentially equivalent larger cluster states formed by fusing the original smaller clusters, as depicted graphically in figure 1.The fusion operation may be achieved as follows.Imagine that we have created two independent clusters that have been redundantly encoded.Suppose that we wish to fuse two of the qubits, one from each cluster.Suppose that these two qubits are encoded using numbers a and b of qubits respectively.Implementing a J-measurement between one member of each cluster qubit's redundant encoding yields the following two possible evolutions, where we have reordered the states so that the measured qubits appear at the very end of each term The J 0 outcome (which occurs with probability 1/4) is fine-after throwing away the residual singlet, it amounts to having simply fused the two qubits into a new cluster qubit with a redundant encoding of a + b − 2 qubits.However, the second term in the expression resulting from the J 1 outcome is undesirable.To project out this piece of the state, we note that this piece has the two qubits which were measured in the |ψ + state, whereas the desired piece has these two qubits in either |00 or |11 .Note also that |ψ + = |+ + − | − − .Thus if single-qubit measurements are performed in the |± basis on these two qubits, and anticorrelated outcomes are obtained (i.e.|+ − or |− + ), then the |ψ + part is projected out, and the desired fused cluster state is obtained.The overall joint probability of obtaining the J 1 outcome followed by these anticorrelated outcomes is 1/4.A failure occurs when the correlated outcome (|+ + or |− − ) is obtained-the overall probability of such a failure is 1/2.Crucially, however; when a failure does occur it is easy to see that the final state of the two clusters is one of depending upon the outcome.Thus, we see that the original cluster states have not been destroyed, all that has happened is that one qubit has been removed from each cluster qubit's redundant encoding.If sufficient qubits remain, the fusion can be reattempted.

Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
Clearly, if we start off with two-qubit cluster states with large enough redundant encodings, then they can be fused with high probability.As in [3,9], simple random walk considerations then show that we efficiently create clusters of arbitrary size by such probabilistic fusion.
We have seen above that we can create a two-qubit cluster |0+ + |1− , which has no redundant encoding-what remains, therefore, is to verify that we can create such clusters with a suitable amount of redundant encoding.If we consider a two-qubit cluster and a GHZ state We readily verify that by applying the fusion procedure outlined above simply creates a suitable redundant encoding.Such unsuccessful fusions need to be discarded-since we envisage such a procedure is being implemented 'offline', this is not an issue.(Similarly, such fusion can be used to probabilistically create large GHZ states from smaller ones.)Note that theorem 1 leaves open the interesting question of whether we actually need three linearly independent supplies of qubit states.It might be that via a smarter encoding it is possible to perform quantum computation with J-measurements and only one or two different types of initial state, although we have not been successful in finding such an encoding.

Proof of property (i).
Provable optimal purification to the largest eigenvector of ρ by using large joint measurements of total angular momentum was described in [10].Here, we perform purification that is not optimal in terms of resources, but uses only the two-qubit J-measurement.
Assume that we have N copies of ρ whose Bloch vector is r 0 .Sort these into N/2 pairs and perform the J-measurement, keeping only those pairs for which J 1 is obtained.The Bloch vector of the reduced state on either side is now r 1 = 4/(3 + r 2 0 ) r 0 .The probability of getting the J 1 outcome is simply P(r 0 ) = (3 + r 2 0 )/4.Imagine we throw away half of these pairs.We are left with NP(r 0 )/2 systems which have the longer Bloch vector r 1 .
How many times must we repeat the process so that the Bloch vector has length r max 1 − ε? Multiple repetitions of the above process yields the recursion relation r n+1 = 4r n /(3 + r 2 n ).This is not easily solved.However, we underestimate the growth of the Bloch vector if we presume it follows the simpler recurrence R n+1 = 4R n /(3 + R n ), that is, r n R n ∀n, with R 0 = r 0 .This latter recurrence is easily solved, yielding From this, we deduce that to obtain R n 1 − , it suffices to take or, more simply, n 3 log(1/ r 0 ) will suffice.Obviously, we will not always be successful in obtaining the J 1 outcome, and so this process will only succeed with some probability.However, as we only need to obtain some constant fault tolerance threshold, the resource requirements for this purification procedure are still constant, and the above arguments are sufficient to show that our purification procedure is efficient enough.Nevertheless, for completeness we may perform an approximate analysis of the overheads involved in our purification method.The probability of success (i.e.obtaining the J 1 outcome) on any given pair is

Conclusions
We have presented a proof that the two-qubit total spin measurement (the 'J-measurement') can be used to affect a cluster-state quantum computation scheme when accompanied by a supply of three linearly independent types of single-qubit mixed state.Our method proceeds by using the J-measurement to purify mixed states, to 'fuse' qubits into cluster states and to simulate single-qubit measurements by 'comparing' them.
Although it has previously been shown that quantum computation can be performed using only two-qubit measurements (see e.g.[12]), our scheme is the only one to use just one dynamical element.The purification methods used in our scheme are somewhat reminiscent of the 'magic states' method put forward by Bravyi and Kitaev [5,13], in which a supply of single-qubit states replaces the need for non-Clifford operations in quantum computation.In the 'magic states' scheme a noisy supply of single-qubit mixed states must first be purified using Clifford operations.In our scheme, we use three types of single-qubit state to avoid the use of all dynamical operations other than the J-measurement, and we must also purify these sources.However, the J-measurement can be used to purify any non-trivial mixed state, whereas Clifford operations can at most be used to purify mixed states outside a particular octahedron.In this sense, it seems that J-measurements are a particularly powerful dynamical operation.
One of the most interesting features of our scheme is the symmetry of the J-measurement.This allows the computation to be 'relational', in that no cartesian reference frame is required at any point in the process.Although it is in principle possible to use the large supplies of qubit states to construct a particular reference frame, the scheme does not require such a frame and in fact does not construct one.For instance, if an unknown randomly fluctuating magnetic field were to rotate all the qubits in the computer in the same manner, it would not affect the computation at all, as the only dynamical element is the J-measurement and this is invariant under such transformations.
There are a few open questions regarding the efficiency of our protocol.In particular, it will be interesting to see how the purification protocols may be optimized, as well as to find out whether the requirement for three types of single qubit can be reduced to just two, or perhaps even one (non-pure) state.

Figure 1 .
Figure 1.Examples of how clusters states can be joined together by 'fusing' qubits from each.