Realization of the WernerHolevo and LandauStreater quantum channels for qutrits on quantum computers
Abstract
We implement the LandauStreater and WernerHolevo 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 pnorm. Our realization of such channels is based on embedding qutrit states into states of two qubits and using singlequbit and twoqubit 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 5qubit and a 20qubit 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 20qubit computer.
I Introduction
A quantum channel (QC) is a completely positive tracepreserving (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 LandauStreater(LS) Landau and Streater (1993) channel and the WernerHolevo(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 pnorm 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 LandauStreater and WernerHolevo channels. In this paper, we follow the experimental study of QC on quantum computers AlvarezRodriguez et al. (2018); Santos (2016); Zhukov et al. (2018); Roffe et al. (2018); Geller (2018); Morris et al. (2019) and implement the LandauStreater and WernerHolevo 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 WernerHolevo and LandauStreater 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 spin1 quantum particles, which interacts with another one. One important direction of this research is the direct experimental study of QC on quantum computers AlvarezRodriguez et al. (2018); Santos (2016); Zhukov et al. (2018); Roffe et al. (2018); Geller (2018).
In this paper, we implement these QCs using only onequbit 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 nonzero 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 WernerHolevo and LandauStreater channels agree well with the theoretical expectations.
Ii Brief description of the LandauStreater channel
The LandauStreater 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 amplitudedamping noise.
We did not use physical twoqubit 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 nonqutrit state has the following matrix:
(6) 
where
Iv Implementation of the LandauStreater channel through the WernerHolevo channel
By definition, the WernerHolevo channel Werner and Holevo (2002) evolves a density matrix() as follows:
(7) 
After the following unitary transformation:
(8) 
we get the LandauStreater 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 WernerHolevo 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 onequbit gates.
As result, we get the following matrix:
(16) 
which is almost equal (some elements might have different phases) to:
(17) 
For effective decomposition into onequbit gates, we replace the Toffoli gate by a quasiToffoli one. It requires fewer CNOT gates (Fig. 2) and has the following matrix:
(18) 
A decomposition into a tensor product of simple gates exist only in 4 out of 8 cases (Fig. 3):
(19) 
(20) 
(21) 
(22) 
Not every CNOT gate can be performed on IBM quantum computers due to missing connections, see connectivity map for the 5qubit machine in Fig. 5 and 20qubit machine in Fig. 6.
On the 5qubit 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 twoqubit CNOT gates is minimal since it has the largest error.
On this machine, we use the configuration implementing the WernerHolevo channel. The sequence of gates for this is shown in figure 7 (, and are a full set of onequbit gates on IBM machines). Qubits encode (channel qubits) and represent (environment qubits).
The transformation (Sec. IV) requires a CNOT gate between the system qubits (). However, the coupling map of the 5qubit 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).
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.
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 WernerHolevo and LandauStreater channels fit well with the theoretical expectations (fidelity of both channels is nearly 0.99).
Vi Choi matrix of the LandauStreater 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 5qubit machine. The connectivity map of 15qubit machines also does not allow a direct implementation. Only on a 20qubit computer (see connectiivty map in Fig 6) we can built the Choi matrix using formula 26. In order to be able to utilize the 5qubit machine, we used another expression, Eqs. (VI) & (30) to reconstruct the Choi matrix .
Using this procedure, we get the Choi matrices for the LandauStreater and WernerHolevo 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).
In contrast to this more involved procedure, we built the Choi matrix straight forwardly using Eq. (26) on a 20qubit 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 LandauStreater (WernerHolevo) 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 LandauStreater and WernerHolevo channels. Experiments were conducted on 5qubit and 20qubit 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. 170200396A, 180200642A and 183720073, 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 20qubit machine were performed on IBM’s quantum computer tokyo at Oak Ridge National Laboratory and gratefully acknowledge access to this machine.References
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