Quantum State and Process Tomography via Adaptive Measurements
We investigate quantum state tomography (QST) for pure states and quantum process tomography (QPT) for unitary channels via adaptive measurements. For a quantum system with a -dimensional Hilbert space, we first propose an adaptive protocol where only measurement outcomes are used to accomplish the QST for all pure states. This idea is then extended to study QPT for unitary channels, where an adaptive unitary process tomography (AUPT) protocol of measurement outcomes is constructed for any unitary channel. We experimentally implement the AUPT protocol in a 2-qubit nuclear magnetic resonance system. We examine the performance of the AUPT protocol when applied to Hadamard gate, gate ( phase gate), and controlled-NOT gate, respectively, as these gates form the universal gate set for quantum information processing purpose. As a comparison, standard QPT is also implemented for each gate. Our experimental results show that the AUPT protocol that reconstructing unitary channels via adaptive measurements significantly reduce the number of experiments required by standard QPT without considerable loss of fidelity.
The problem of how many measurements are needed to determine a wave function of a quantum system is a nontrivial task even in principle, and has attracted considerable attention over the history of the subject. Originally raised by Pauli in 1933, the problem was framed as whether the probability distribution of position and momentum is enough to determine the wave function Pauli (1933). Subsequently, various versions of the problem and many different approaches have been explored Weigert (1992); Amiet and Weigert (1999).
For a system of finite dimension with Hilbert space , a normalized pure state is specified by real parameters. To measure any observable on the ensemble of identical copies of the states , the expectation is returned. In order to determine an arbitrary , at least such observables need to be measured.
The development of quantum information science has shed new light on the problem Flammia et al. (2005); Finkelstein (2004); Gross et al. (2010); Cramer et al. (2010); Heinosaari et al. (2013); Chen et al. (2013); Baldwin et al. (2015); Li et al. (2015); Ma et al. (2016); Xin et al. (2016); Carmeli et al. (2016), which can be rephrased by quantum state tomography (QST) for pure states in the language of quantum information. In particular, the precise meaning of the word ‘determine’ is clarified, where two important scenarios are considered Chen et al. (2013). The first scenario is whether the measurement results uniquely determine the pure state among all pure states (UDP, i.e. no other pure states can give the same measurement result) or among all states (UDA, i.e. no other states, pure or mixed, can give the same measurement result). The latter is an arguably stronger requirement and gaps are found between the number of measurements needed for UDP and UDA. The second scenario is whether the measurement results determine (UDP or UDA) all pure states (i.e. any state can be reconstructed unambiguously) or just generic pure states (i.e. almost all pure states are determined except a set of states that are of measure zero). The former is an arguably stronger requirement and gaps are found between the number of measurements needed for all pure states and generic pure states Chen et al. (2013). In columns and of Table 1, we summarize the best known number of measurements needed for UDP/UDA for all states in the row of starting with “All” and for generic states in the row of starting with “Generic”.
Now we naturally extend the above problem of QST for pure states to quantum process tomography (QPT) for unitary channels. QPT for unitary channels has the goal of determining an unknown unitary operation. A unitary operation has real parameters, compared to a general quantum channel on a -dimensional system that have real parameters. In Gutoski and Johnston (2014), it is shown that measurements are sufficient to identify a unitary channel among unitary channels, non-adaptively. Their method is based on the state tomography of the corresponding Choi matrix of the unitary channel. Ref. Baldwin et al. (2014) provides a nonadaptive method of unitary tomography using measurements, whereas it does not works for all unitary channels but for almost all unitary channels (i.e. works for ‘generic channels’). Their method is based on the fact that each column of the unitary matrix can be determined by QST for an input state that is a computational basis state, and the relative phases between any two columns of can be further determined by QST for some input states that are superpositions of computational basis states. We summarize these results in column of Table 1.
All the above-mentioned protocols for either QST or QPT are non-adaptive, that is, the observables to be measured are fixed once chosen. One can also consider adaptive measurement by allowing measurements that are determined by the results of the previous measurements. There has been such trials along this direction, and a measurements protocol via adaptive measurements are discussed in Goyeneche et al. (2014), for UDA all pure states. One important open question is what are the minimum number of measurements needed for QST of all pure states, and for QPT of all unitary channels.
In this work, we study QST for pure states and QPT for unitary channels, using adaptive measurements. For QST, we show that measurements are enough to UDA (hence UDP) all pure states, by adaptive measurements. This is a significant improvement over the lower bound for UDP using non-adaptive measurements Heinosaari et al. (2013). We then further apply our protocol to study QPT of unitary channels, and show that measurements are sufficient to reconstruct all unitary channels when adaptive scheme is allowed.
|All (Nonadaptive)||Heinosaari et al. (2013)||Chen et al. (2013)||Gutoski and Johnston (2014)|
|Generic (Nonadaptive)||Finkelstein (2004)||Baldwin et al. (2015)||Baldwin et al. (2014)|
We organize our paper as follows: in Sec. II, we discuss an adaptive protocol that UDA (hence UDP) for all pure states with measuring observables; we then apply this protocol on QPT of unitary channels by measuring observables. In Sec. III, we discuss an adaptive experimental protocol of QPT for two-qubit unitary channels. In Sec. IV, we implement the experimental protocol in a two-qubit NMR system. Our experimental results are discussed in Sec. V, followed by a brief conclusion in Sec. VI.
Ii Adaptive protocols for quantum state and process tomography
A Adaptive Pure State Tomography
In this subsection, we propose an adaptive pure state tomography (APST) protocol for -dimensional pure states using at most observables.
The state space we considered is spanned by orthogonal basis . Suppose the quantum state is
The goal of tomography is to obtain all ’s for , and the APST protocol is given as follows:
Step 1. Measure using measurements sequentially until is non-zero, where . The goal is to find the smallest such that . Hence this step costs measurements, and the state becomes
where the summation starts from now. Without loss of generality, we assume that is real since the global phase of a quantum state is ignorable.
Step 2. Measure using measurements for all with Hermitian and . The goal of this step is to obtain for all by employing the coherence between and . This step costs measurements.
In total, the number of measurements is which is no more than , depending on when we have measured the non-zero for the first time. In terms of density matrix, our protocol actually provides the ()-th row of .
In the following, we analyze this protocol and show that it indeed accomplishes the task of QST for pure states. In other words, one can compute each according to the outcomes of this protocol.
We first show that it is UDP. After step 1, we know that . After step 2, we have
As we have assumed that is real in step 1, it is obvious that for all . Therefore, we can calculate the exact value of since we know the non-zero and from our measurements. It means we have the complete information of if we know it is pure.
Next we prove that this APST protocol is not only UDP, but also UDA. To see this, we need to show that if another quantum state which gives the same results as , can only be .
Assume there exists another quantum state that has the same measurement results compared to . So for , we have
For , we have
In other words, our protocol actually outputs the first non-zero row of , which is the -th row. This row of equals to the -th row of .
As is semi-definite positive, we can suppose
with unnormalized . According to Eq. (4) and the semi-definite positive property of , we know that the first rows of are all zero, namely, for all .
Without loss of generality, we can assume that and for all . This property helps us to show . To achieve such a decomposition, we first observe that
Apply this on by choosing appropriately, we can always achieve . Employing this argument recursively on , we can similarly have for all . Then, the -th row of are all zero.
According to , we observe that the -th row of equals to the -th row of . Thus, the -th row of equals to the -th row of . Therefore, equals to up to a global phase, which means . Thus,
That is , and
This verifies our claim that our APST protocol is UDA and uses only measurements.
B Adaptive Unitary Process Tomography
In this subsection, the idea of APST is generalized to deal with the adaptive unitary process tomography (AUPT). We notice that the unitary map can be written as a transformation from the orthonormal basis to its image basis ,
The task of QPT for a unitary map is to fully characterize the basis and the relative phases , and our AUPT protocol consists of steps as follows,
Step . Implement QST for . We use the APST protocol in the previous subsection to characterize . This step costs at most measurements.
Step . Implement QST for
The goal of this step is to tomography and to obtain the relative phase between and simultaneously. This can be done by obtaining
using our APST protocol, so that we can construct . To see this, we notice that . Observe that the inner product of and is . This indicates that the phase information of is obtained. Then, the information of is obtained.
Moreover, we observe that
has the only solution that . That implies that the information of the relative phase between and is obtained completely. We choose a basis
with . and then apply this basis using step 2 in the APST protocol to obtain .
This step costs measurements since we already know the amplitude of is .
Step . Implement QST for
The goal of this step is to tomography and obtain the relative phase between and simultaneously. The procedure to obtain is similar to step 2 by choosing a basis
with for all , and applying it on the APST protocol.
This step costs measurements, since we already know the amplitude of is for state , and the amplitudes of are all zero.
The above steps keep going until step , in which two measurements are required and the complete information of is obtained from the outcomes of the steps. This AUPT protocol thus uses measurements.
Iii Experimental protocol
In this section, we show how to apply our AUPT protocol to characterize unitary channels (as discussed in Sec. B) in a 2-qubit NMR system, and its complexity, i.e. the number of measurements in terms of Pauli operators. As a comparison, we also briefly review how to implement a standard QPT and the complexity. The extension of our protocol to arbitrary sizes is straightforward.
A Standard QPT
First let us recall the procedure of a 2-qubit standard QPT. Suppose is the -qubit unitary gate that we want to implement in practice. Due to the inevitable experiment errors, the real quantum channel in the laboratory is no longer unitary, but still some completely positive trace-preserving (CPTP) operation, denoted by . In NMR and most of ensemble systems, it is convenient to prepare and measure Pauli observables, hence we use the representation of Pauli observables to describe such a 2-qubit channel . Note that this description is equivalent to the Choi matrix representation and they can be easily transformed to each other Chow et al. (2012).
Therefore, can be written in the way of mapping Pauli group to Pauli group so that
where all elements () are real. To reconstruct in NMR, we firstly prepare the initial state as , and then apply on it. The output state is thus . By doing a full state tomography in 15 experiments, i.e. measuring each (, since can only be computed via the normalization condition), we can obtain the first column of . To fully characterize , the above procedure needs to be repeated by times, with each time preparing a distinct Pauli input state out of . So the total number of experiments to reconstruct a 2-qubit channel is .
If we assume is still unitary when applied in practice, the total number of experiments can be reduced significantly. Due to the experiment errors, let us denote as the real channel, which is still unitary but deviates from the desired . As unitary operators do not change the purities when applied on quantum states, it is convenient to consider the map from pure states to pure states. Explicitly, the map of can be written as
where the elements in are all complex numbers. Similarly to standard QPT, in experiment we firstly prepare and then apply . The output quantum state is still pure since
Now the problem of characterizing a unitary channel converts to the QST of a pure state. First, we can use three measurements of the diagonal elements combined with the normalization condition to get and . Then we need to measure the relative phase between all the ’s. Specifically, we pick out the maximal and set its phase as zero. Without loss of generality, assume is the largest one and set it as reference. To measure, for instance, the relative phase between and , is equivalent to extracting the phase between and in experiment, which requires two measurements of and on the second qubit. Analogously, the relative phase and can be measured with four more experiments. Therefore, the total number of experiments to extract the values of in the first column is nine, with three for moduli and six for relative phases. As contains four columns, this procedure is repeated by four times that necessitates 36 experiments, by preparing the input state as , , and , respectively.
However, the above procedure cannot provide the information of the relative phases between columns, as we have set the phase of the maximal element in each column as zero, but quantum mechanics merely allows one ignorable global phase. So the next step is to determine these relative phases between columns. Without loss of generality, assume is real. To measure the relative phase between the column and column, one can adopt the idea of interferometers. Explicitly, prepare the superposition as the input state and apply , so that
In practice, we can measure the relative phase between and via two experiments, and the desired can be obtained by solving the following equation
where means the relative phase between two complex numbers and , and all and values have been obtained in the last step. Similarly, the relative phases and can be obtained through preparing and , applying , and measuring the corresponding phases. This step thus consists of six experiments to acquire three relative phases between columns in .
In total, we need experiments to characterize a 2-qubit unitary process via the AUPT protocol, significantly less than the standard QPT which requires 210 experiments.
Iv Experimental implementation in NMR
Now we turn to the experimental demonstration of the AUPT protocol for 2-qubit unitary gates in the NMR system. Five elementary gates , , , and are chosen due to the fact that any 2-qubit quantum circuit can be decomposed into these five gates in arbitrary accuracy Nielsen and Chuang (2010). is the identity operator, and the Hadamard gate , gate and controlled-not gate are
The experiments are carried out at room temperature on a Bruker AV-400 spectrometer (9.4 T). The physical system is carbon-13 enriched chloroform (CHCL) dissolved in deuterated acetone. One H nucleus and one C nucleus of spin-1/2 are encoded as qubit and qubit , respectively. The molecular structure and relevant parameters are shown in Fig. 1. In the rotating frame, the internal Hamiltonian of the system can be written as
where is the scalar coupling strength between the two nuclei.
In NMR system, the thermal equilibrium state is a mixed state , where is the identity matrix, is the polarization, and is the deviation density matrix. The coefficients of and come from the fact that the gyromagnetic ratio of H is four times larger than C. Note that the dominant identity part is invariant under unital propagators, so we only consider the deviation part in experiment.
As reference, we firstly implement the standard QPT in experiment. The creation of all Pauli input states are realized by single-qubit rotations, free evolutions under the internal Hamiltonian, and -gradient field pulses (to crush the unwanted non-zeroth coherence which is necessary in creating ) from the thermal equilibrium state. Then we apply the five gates to these Pauli input states via the following pulse sequences (pulses applied from right to left)
The notation represents a single-qubit rotation on qubit along the -axis with the rotating angle , and represents the free evolution under the internal Hamiltonian in Eq. (15) with time . The -rotations in and can be decomposed by the formula . Note that -rotations in NMR can also be realized more precisely by virtually varying the reference frame Ryan et al. (2008), but this approach is not used in this experiment as it is better to apply imperfect pulses in order to address the stability of the AUPT protocol. Finally, by doing full QST, we reconstruct each gate in terms of an imperfect quantum channel as described in Eq. (8).
For the AUPT protocol, starting from the thermal equilibrium state, we firstly create the pseudo-pure state (PPS)
using the spatial average technique Cory et al. (1997); Lu et al. (2011). The other states , and are created from by rotations. After applying one of the five gates, we measure the module of each element in in Eq. (11) by standard tomography of diagonal elements in NMR. The relative phases within a given column correspond to the phases of single coherent terms, which is straightforward to read out in NMR as the spectrometer uses quadrature detection. In fact, all relative phases within one column for all five gates can be obtained in this way.
Next we need the relative phases between columns for each gate. We initialize , and as the input states by applying rotations on the PPS state, which enables the reconstruction of all relative phases between columns for , , and . However, is an exception. The application of to generates double quantum coherence which cannot be directly read out in NMR. The solution is to apply another gate before detection to evolve double coherence back to single coherence, which may roughly double the error in . Till now, we have successfully implemented the AUPT protocol for all five gates, and characterized each in Eq. (9) individually.
V results and discussion
The reconstructions of each gate , , , and via the standard QPT and AUPT protocols, as well as the theoretical results, are all shown in Fig. 2. The five rows show the five gates, and the left, middle and right column are the theoretical, standard QPT, and AUPT results, respectively. Each subfigure shows the complete information of the target channel in the Pauli representation, as shown by the 16-by-16 matrix in Eq. (8). Note that the AUPT results (the right column) are initially obtained via Eq. (11) which is 4-by-4, and then converted to their equivalent 16-by-16 matrices in Eq. (8) for fair comparisons with the other results. In each subfigure, the -axis describes the input state in Eq. (8), and -axis describes the output state in the Pauli basis after applying the current channel. For example, the first column in each subfigure shows that when applying the channel to the input state , what the coefficients of the output state in Pauli basis are. From Fig. 2, we see that the standard QPT results are closer to the theoretical predictions than the AUPT results.
To describe how closely that the practical channel approximates the theoretical channel which is unitary in our case, one can use the value of diamond norm Benenti and Strini (2010) or average fidelity. Here we use the average fidelity between two channels, which is defined as
where is the unitarily invariant distribution of pure states known as Fubini-Study measure Emerson et al. (2005). For simplicity, we randomly sample 1000 ’s from the 2-qubit pure state space, and replace the integral in Eq. (18) by the sum (with some normalization). The calculated average fidelities of the standard QPT protocol and AUPT protocol , both compared with the theoretical results , are shown in Table 2 for all the five gates. To get each average fidelity and its uncertainty, we randomly sample 1000 2-qubit pure states to get one value via Eq. (18) and repeat this procedure for 100 times. The average fidelity and uncertainty are defined as the mean and standard deviation of the 100 repetitions. The uncertainty for each gate is very small, which means 1000 samples are sufficient to estimate the average fidelity with a high precision.
Now let us discuss the error sources in two aspects. First of all, both of the standard QPT and AUPT results suffer the decoherence effect, imperfection of pulses, and state preparation and measurement (SPAM) errors. The decoherence is almost ignorable, as the gate implementation time is less than 3 ms, much shorter than the relaxation time scales which are at least 350 ms as shown in Fig. 1. The imperfection of pulses such as over-rotation and under-rotation induce the SPAM errors, as well as the target gate infidelity. Just to clarify, it is hard for either the standard QPT or the AUPT protocol to distinguish the wanted gate error from the SPAM errors, but these two protocols both provide complete information of an unknown quantum channel. In contrast, the randomized benchmarking protocol Emerson et al. (2007) enables the separation of the gate error rate from the SPAM errors, but fruitless in fully characterizing the quantum channel.
Secondly, in Table 2 the AUPT results are worse than the standard QPT results (except , for which we think the fluctuations in the SPAM error dominate the infidelity, and make it singular). The reason can be attributed to two factors. On one hand, the AUPT protocol is adaptive, that the next measurement relies on the previous one. It enables the propagation and amplification of the error to the latter experiments from the earlier experiments. On the other hand, to measure the relative phase via Eq. (12), we need to know the modules for each element and choose the single coherence — the only term that can be observed directly in NMR. An extreme case is the gate, that we have to apply it twice in order to evolve double coherence back to single coherence and observe the relative phase. That is why the AUPT result of is much worse than the case of standard QPT. Therefore, we conclude that AUPT indeed improves the efficiency significantly in characterizing an unknown quantum channel experimentally by assuming it is unitary, whereas it does have some drawbacks such as the two issues mentioned above.
In summary, we studied the quantum state tomography and unitary channel tomography via adaptive measurements. We showed that adaptive measurements can reduce the number of measurements when compared to non-adaptive measurements. In particular, we proved that pure state tomography can be accomplished using measurements. By employing this idea, we demonstrated that measurements are sufficient to reconstruct a unitary process when the adaptive scheme is allowed.
Additionally, we implement our AUPT protocol for the universal gate set of quantum computing in a 2-qubit NMR system. Our results show that for local gates such as Hadamard and ( phase) gates, high fidelities can be achieved using the AUPT protocol. For two-body gate such as the CNOT gate, the fidelity drops by some amount due to the accumulation of the errors in adaptively measuring the relative phases. Nevertheless, the AUPT protocol is still a useful tool in characterizing the unitary channels as it allows a significant reduction in terms of the required experiments, in particular for the local unitary channels.
Acknowledgements.We are grateful to the following funding sources: NSERC (N.Y., D.L., D.K., and B.Z.); CIFAR (B.Z.); National Natural Science Foundation of China under Grants No. 11175094 and No. 91221205 (K.L., and T.X.); National Basic Research Program of China under Grant No. 2015CB921002 (K.L., and T.X.). H.W., X.P., and J.D. would like to thank the following funding sources: NKBRP (2013CB921800 and 2014CB848700), the National Science Fund for Distinguished Young Scholars (11425523), NSFC (11375167, 11227901 and 91021005).
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