Realization of the Werner-Holevo and Landau-Streater quantum channels for qutrits on quantum computers

Realization of the Werner-Holevo and Landau-Streater quantum channels for qutrits on quantum computers

A. I. Pakhomchik Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia    I. Feshchenko Skolkovo Institute of Science and Technology, 5 Nobelya str., Mozhajskij region, Moscow, 121205, Russia Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia    A. Glatz Materials Science Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA    V. M. Vinokur Materials Science Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA    A. V. Lebedev Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia    S. N. Filippov Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina St. 8, Moscow 119991, Russia Institute of Physics and Technology of the Russian Academy of Sciences,Nakhimovskii Pr. 34, Moscow 117218, Russia    G. B. Lesovik Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia
Abstract

We implement the Landau-Streater and Werner-Holevo quantum channels for qutrits on IBM quantum computers. These quantum channels correspond to a specific interaction between a qutrit with its environment, which results in the globally unitarily covariant qutrit transformation violating the multiplicativity of the maximal p-norm. Our realization of such channels is based on embedding qutrit states into states of two qubits and using single-qubit and two-qubit CNOT gates to implement the specific interaction. We exploit the standard quantum gates only, so the developed algorithm can be executed on any quantum computer. We run our algorithm on a 5-qubit and a 20-qubit computer as well as on a simulator. We quantify the quality of the implemented channels by comparing their action on different input states with theoretical predictions. The overall efficiency is quantified via the fidelity between the theoretical and experimental Choi states, with the latter being physically implemented on the 20-qubit computer.

I Introduction

A quantum channel (QC) is a completely positive trace-preserving (CPTP) map, , between operators defined on Hilbert space . In particular, a quantum channel conveniently describes the transformation of a density matrix, , that interacts with an environmentNielsen and Chuang (2010); Breuer and Petruccione (2002); Holevo (2012). Unitary evolution of the closed system is an example of QC that preserves the maximally mixed state i.e. . It is shownLandau and Streater (1993) that any open quantum dynamics of a qubit that preserves the maximally mixed state is essentially random unitary , however, in greater dimensions this is not the case. Namely, for a qutrit (d=3) there exists a quantum channel such that but is not random unitary Landau and Streater (1993); Audenaert and Scheel (2008); Mendl and Wolf (2009). It is the Landau-Streater(LS) Landau and Streater (1993) channel and the Werner-Holevo(WH) Werner and Holevo (2002) channel that satisfy this property. Moreover, such qutrit channels are extremal, exhibit the global unitary covariance, and violate the multiplicativity of the maximal p-norm Werner and Holevo (2002). Quantum informational properties of these channels are reviewed Filippov and Kuzhamuratova (2019). It is of great interest to physically implement the specific interaction of a qutrit with environment that would produce the Landau-Streater and Werner-Holevo channels. In this paper, we follow the experimental study of QC on quantum computers Alvarez-Rodriguez et al. (2018); Santos (2016); Zhukov et al. (2018); Roffe et al. (2018); Geller (2018); Morris et al. (2019) and implement the Landau-Streater and Werner-Holevo quantum channels for qutrits on IBM quantum computers with 5 and 20 qubits.

For example, the time evolution of a quantum system can be described by the map with the unitary time evolution operator ( is the Hamiltonian), is a CPTP map, i.e. a quantum channel in the Schrödinger picture. Quantum channels have a special subset of channels that are called unital channels, which preserve the identity matrix, . The Werner-Holevo and Landau-Streater quantum qutrit channels which we study in this work are unital. These quantum channels have several interesting properties. In particular, they are not random unitary, meaning that they do not change the chaotic state with maximum entropy and cannot be presented as , where are unitary operators. They have been subject of extensive recent study Filippov and Kuzhamuratova (2019); Audenaert and Scheel (2008); Mendl and Wolf (2009). In terms of applications in quantum physics, the QCs can be used to describe the time evolution of a spin-1 quantum particles, which interacts with another one. One important direction of this research is the direct experimental study of QC on quantum computers Alvarez-Rodriguez et al. (2018); Santos (2016); Zhukov et al. (2018); Roffe et al. (2018); Geller (2018).

In this paper, we implement these QCs using only one-qubit and CNOT gates. As shown in Nielsen and Chuang (2010), any unitary operation can be approximated using these gates up to arbitrary accuracy. We represented a qutrit, a quantum state being equivalent to a particle with spin , as two qubits ignoring the highest energy level . Since the used quantum computers, IBM QUANTUM EXPERIENCEIBM (2016a), are not ideal and subject to a noise, the final density matrix may have a non-zero probability of being in the ignored state. For example, a density matrix can transform in the following one: . We discard the last term and normalize it by the trace , i.e., we consider only a qutrit part of the density matrix ().

As result, we build the Choi matrix corresponding to the QCs from the output density matrices by using different input states. According to the Choi–Jamiołkowski isomorphism, the Choi matrix contains all information about the considered channel. Finally, we compared the obtained matrices with theoretical expectations. The latter is obtained by using a simulator of an ideal quantum computer, which is free of measurement errors and other types of errors related to the coupling to the environment. As the main result, we find that all the tomography experiments for the Werner-Holevo and Landau-Streater channels agree well with the theoretical expectations.

Ii Brief description of the Landau-Streater channel

The Landau-Streater channel acts on on the state as follows

(1)

and is defined through the generators acting on a -dimensional Hilbert space for a spin- particle.

(2)

In a previous paper Filippov and Kuzhamuratova (2019) the Stinespring representation for the LS channel was derived and relevant matrices and calculated such that they satisfy the following equation:

(3)

These are given by

(4)

and

(5)

Iii Emulation of qutrits by qubits

The Hilbert space of two qutrits (where ) has basis states and the Hilbert space of qubits has basis states. Therefore, we need qubits for the 9 states of two qutrits, which are sufficient to encode a system qutrit that is transformed by the channel and a separate environment qutrit. In other words, one qutrit uses one pair of qubits, and the other qutrit uses another.

For encoding the three logical states , , and of a qutrit we use the states , , and , respectively. This choice is related to the fact that the quantum computer is not ideal, meaning that excited states return over time into the ground state due to amplitude-damping noise.

We did not use physical two-qubit state , because it has the highest energy and, therefore, the shortest relaxation time into other states. It can decay into the two states: and , which in their turn can only return to ground state . In summary, the algorithm that transforms states by and acts trivially on the other non-qutrit state has the following matrix:

(6)

where

Iv Implementation of the Landau-Streater channel through the Werner-Holevo channel

By definition, the Werner-Holevo channel Werner and Holevo (2002) evolves a density matrix() as follows:

(7)

After the following unitary transformation:

(8)

we get the Landau-Streater channel accordingly:

(9)

The WH channel can be represented as follows:

(10)

where

(11)

and

(12)

The only requirement for the and blocks is that is unitary.

With the selected method of encoding logical states of a qutrit by two qubits. we implemented the transformation (sec. IV) as follows:

(13)

This can be decomposed into elementary gates in this way:

(14)

In the matrix emulation, Sec. IV, by qubits that was described in Sec. III, the impact on the unused state (which does not encode a logical qutrit) could be any if it preserves the unitarity of the matrix. Taking advantage of this condition, we represent the matrix for Werner-Holevo channel as:

(15)

The columns of are arbitrary with only one condition: the matrix must satisfy the unitary constraint. Using the Toffoli and CNOT gates, we make lines permutations, , (Fig. 1) of the matrix such that its absolute values are equal to a tensor product of one-qubit gates.

As result, we get the following matrix:

(16)

which is almost equal (some elements might have different phases) to:

(17)
Figure 1: Algorithm for line permutations. Qubits and encode the environment qutrit and qubits and encode the channel-transformed spin-1 particle.

For effective decomposition into one-qubit gates, we replace the Toffoli gate by a quasi-Toffoli one. It requires fewer CNOT gates (Fig. 2) and has the following matrix:

(18)
Figure 2: Two different implementations of a quasi-Toffoli gate.

A decomposition into a tensor product of simple gates exist only in 4 out of 8 cases (Fig. 3):

(19)
(20)
(21)
(22)
Figure 3: Implementations of the algorithm shown in Fig. 1, using the CNOT decomposition of quasi-Toffoli gates. Here .

Not every CNOT gate can be performed on IBM quantum computers due to missing connections, see connectivity map for the 5-qubit machine in Fig. 5 and 20-qubit machine in Fig. 6.

Figure 4: Coupling map of two-qubit gates on a 5-qubit machine
Figure 5: Swap algorithm for the target and control qubits of a CNOT gate.
Figure 6: Connectivity map of IBM’s 20-qubit machine tokyo. The (yellow) highlighted qubits where used to implement the quantum channels.

On the 5-qubit machine each couple of qubits in the coupling map has only one target and control qubit. Therefore, if we need to swap the control and target qubit, we need to use the method shown in Fig. 5 adding Hadamard gates. We chose a configuration from the figure 1 such that the number of two-qubit CNOT gates is minimal since it has the largest error.

On this machine, we use the configuration implementing the Werner-Holevo channel. The sequence of gates for this is shown in figure 7 (, and are a full set of one-qubit gates on IBM machines). Qubits encode (channel qubits) and represent (environment qubits).

Figure 7: Implementation of the configuration (Fig. 1) on a 5-quibit machine, determined by its coupling map.

The transformation (Sec. IV) requires a CNOT gate between the system qubits (). However, the coupling map of the 5-qubit machine does not allow to place a CNOT operator there. Therefore, we have to use three CNOT gates instead of a single one and place them in accordance with the coupling map (Fig. 5).

Figure 8: Realization of Landau-Streater based on the Werner-Holevo channel.

V Measurements

In order to predict the final quantum state for an arbitrary initial state, we performed the transformation of each basis density matrix per channel. Explicitly, we use the following 9 matrices as inputs:

(23)

After the transformation by the channel, we perform a tomography Paris and Rehá cek (2004) of the system qutrit. Knowing these values, we can reconstruct a channel using its linear properties:

(24)

Any of basis states can be prepared using unitary transformations, see Fig. 9.

Figure 9: Preparation of initial states. If all qubits were connected to each other on the 5-quibit machine, only one CNOT gate would be required for and , but given the current coupling map, we need to use 3 CNOTs.

In practice, IBM has implemented the tomography algorithm in their python library (QISKitIBM (2016b)). We use it instead of our own realization.

As a quality test, we used the following expression to calculate the fidelity Jozsa (1994) between the theoretical and experimental output density matrix for same inputs:

(25)

where .

The QISKitIBM (2016b) toolkit also includes a simulator of an ideal quantum computer, which has no measurement or environment coupling errors. We this to obtain the theoretically expected results. Results of all tomography experiments on simulator for the Werner-Holevo and Landau-Streater channels fit well with the theoretical expectations (fidelity of both channels is nearly 0.99).

Vi Choi matrix of the Landau-Streater channel and comparison to theoretical expectations

Any quantum channel acting on Hilbert space is connected to a linear map in (Choi–Jamiołkowski isomorphism). The duality between channels and states refers to the map:

(26)

where is the quantum channel, the identity channel (), and .

(27)

This formula allows one to restore the Choi matrix by measuring final states. As one can see, products might not be the density operators (, if ).

But each one can be represented as a linear combination of physical matrices (). For convenience, denotes . In case the dimension is equal to n = 3, we get following expression:

(28)

where . Then, the Choi matrix can be expressed as:

(29)

For illustrative purposes, we present it as a block matrix:

(30)

Consequently, in order to build the Choi matrix , it is sufficient to know how the channel acts on the basis matrices .

Here, we use four qubits for channel implementation. If one use the previous expression (26), one needs an extra qutrit (plus 2 qubits). Overall, this requires 6 qubits, thus one cannot calculate the Choi matrix directly on a 5-qubit machine. The connectivity map of 15-qubit machines also does not allow a direct implementation. Only on a 20-qubit computer (see connectiivty map in Fig 6) we can built the Choi matrix using formula 26. In order to be able to utilize the 5-qubit machine, we used another expression, Eqs. (VI) & (30) to reconstruct the Choi matrix .

Figure 10: Fidelity by pairs of density matrices. Every pair stick contains maximum, minimum, and average (a point on a stick) The dashed line represents the fidelity between theoretical and experimental Choi matrices .

Using this procedure, we get the Choi matrices for the Landau-Streater and Werner-Holevo channels. We compared them with the theoretical expectations, and obtained the values 0.406 and 0.419 for the fidelity (), respectively.

Since the Choi matrix contains all information about the channel map, we can reconstruct it according to the following expression:

(31)

In figure 10, we show the fidelity between theoretical and experimental results for both channels. For each basis pair of the density matrices, we calculated the fidelity of using Eq. (31). Each bar for the pairs shows maximum, minimum, and average (point on the bar).

Figure 11: Preparation of a qutrit superposition state . 1.910 is the approximate value of
Figure 12: Fidelity by pairs of density matrices. Each bar shows maximum, minimum, and average (point). The dashed line represents the fidelity between theoretical and experimental Choi matrices .

In contrast to this more involved procedure, we built the Choi matrix straight forwardly using Eq. (26) on a 20-qubit computer (IBM machine tokyo). Its connectivity map is shown in Fig. 6, where we used the highlighted 6 qubits, which are sufficient for the required placement of CNOT gates. According to this expression, we need to prepare the state , for which we first create the state on our system qubits using the scheme of Fig. 11. Then, we place CNOT gates between system qubits and auxiliary qubits such that the control qubit is a system qubit and the target qubit is auxiliary. After these steps, we get the state and apply the Landau-Streater (Werner-Holevo) channel transformations on this state and performed the tomography. The results of these experiments is shown in Fig. 12.

Vii Conclusion

We developed and implemented algorithms for the Landau-Streater and Werner-Holevo channels. Experiments were conducted on 5-qubit and 20-qubit quantum computers. The large errors encountered in the calculations can be mostly attributed to the CNOT gate errors, which have the largest error rate and are extensively used in the algorithm. In the future it might be possible to reduce the number of used CNOT gates and thus to increase the overall fidelity of the experiments.

Acknowledgements.
The research was supported by the Government of the Russian Federation (Agreement 05.Y09.21.0018), by the RFBR Grants No. 17-02-00396A, 18-02-00642A and 18-37-20073, Foundation for the Advancement of Theoretical Physics and Mathematics ”BASIS”, the Ministry of Education and Science of the Russian Federation 16.7162.2017/8.9. The work at Argonne (A.G. and V.M.V) was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. The experiments on the 20-qubit machine were performed on IBM’s quantum computer tokyo at Oak Ridge National Laboratory and gratefully acknowledge access to this machine.

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