Superdense Coding Beyond the Pauli Gates

Proceedings of Student-Faculty Research Day, CSIS, Pace University, May 8th, 2020 SuperDense Coding Beyond the Pauli Gates

Lewis Westfall Pace University, Pleasantville NY, 10570, USA, [email protected]

Abstract—Superdense coding is the quantum ability to trans- she will use to encode this message information to her qubit. mit two classical bits, four states, via one quantum bit (qubit) of These states are can be used to convey any four states. a pair of entangled qubits. This is almost always demonstrated + 1 using the Φ Bell state, √ (|00i + |11i) and the Pauli gates to Because of the entanglement, the single qubit operation, 2 encode the data. This paper investigates the feasibility of using the represented by a 2x2 matrix, is insufficient to describe the u3 gate to encode a fifth state, first by analyzing the linear algebra effects the gate has on the two entangled qubits, since the that describes the process and then by running the process on operation also affects the second qubit. To understand the full the IBM Quantum Experience quantum computer. It is shown effects of her operation, the gate that Alice applies must be the that this does indeed work, making it possible to communicate tensor product of her gate and the identity (I) gate to generate more than four states using a single entangled qubit. a 4x4 operational matrix . A single qubit is represented by a Keywords—Superdense Coding; Quantum Computing; Classical 1x2 column matrix. Because the two qubits are entangled, they Bits; QuBits; Quantum Coding; Entanglement; IBM; Q Experi- must be represented by a 1x4 column matrix and this requires ence; H Gate; CNot, Quantum Channel; Bell State a 4x4 matrix to describe the operation. For instance, if we were to apply a Pauli X (X) gate we would take the tensor product I.INTRODUCTION of the X gate and the identity (I) gate as described below. Superdense coding is the ability for one of an entangled pair of qubits to carry 2 bits of information which is more  0 0 1 0  than a single classical bit allows [1]. This is possible because 0 1 1 0 0 0 0 1 ⊗ =   of quantum entanglement, which connects the quantum state 1 0 0 1  1 0 0 0  of one qubit with the quantum state of another qubit [2]. The 0 1 0 0 use of superdense coding by entanglement is a quantum cryptography method of great promise to defeat any eavesdropper Although the qubits are entangled, while the qubits are in [3]. Frequently the example is of one person, Alice is the usual the possession of their original owner, only Alice can perform person given, sending two classical bits of information in one gate operations on her qubit, and only Bob can perform gate qubit to another person, usually Bob. Frequently the example operations on his qubit. Remembering that any operation on is of Alice telling Bob that the weather is at her location, but either of the qubits affect the pair. Once Alice’s qubit has been actually the definition of the four states can be anything Alice encoded by operations of the quantum gates, it is transmitted and Bob agree to ahead of time. The process starts when a to Bob. third party, call him Oscar for originator, starting with two separate qubits, entangles the pair by applying a Hadamard Bob, with Alice’s qubit now in his possession, as well as (H) gate to the first qubit and then a Controlled Not (CNOT) his own qubit, can perform quantum gate operations on Alice’s gate to both qubits, where the first qubit is the control and qubit to decode the message in it. Bob first applies a CNOT the second qubit is the target. This entanglement produces the operation to both entangled qubits, where Alice’s qubit is the |Φ+i Bell state [4], √1 (|00i + |11i). It is important in this control and Bob’s qubit is the target. Applying the CNOT to an 2 exercise to remember that quantum gates are reversible, unlike entangled pair of qubits causes them to become disentangled classical gates. Whatever a CNOT gate does a second CNOT and break into two independent qubits. Bob measures the gate undoes. The CNOT can entangle, and disentangle. One second qubit, the formerly target qubit that was Bob’s qubit. of the pair of entangled qubit is given to Alice while the other This measurement ends the qubit’s active quantum life, but it is given to Bob. The qubits of Alice and Bob will remain reveals half of the state encoded in the pair. entangled, even though they are physically separated and one Bob then applies an H gate to Alice’s original qubit, the of the qubits is subjected to more gates than the other qubit. first qubit, the former control qubit of the now disentangled State Quantum Gate pair. The H gate operation extracts the second bit of informa- 00 I tion in the message specifying the state. When the qubit is 01 Z measured the binary code that reveals the message that Alice 10 X 11 ZX encoded in her qubit is displayed. The measurements end the process [5]. TABLE I: Alice’s States and Quantum Gates to Encode the States Circuit 1 shows the complete quantum circuit diagram for Alice using the X gate to encode the 10 state. We see the Hadamard and CNOT gates used by Oscar to entangle, Table I shows the options Alice has to send the message to the X gate used by Alice to encode the message, and finally Bob, the binary code for that message, and the quantum gates the CNOT and Hadamard gates used by Bob to extract the 1 message. researchers did not use a real quantum computer or a quantum simulator to test their ideas [8] • • (1) H X H Even when the researchers on a government funded study visited the engineers on the site of the real IBM quantum computers and made note of that in their paper, they still did not conduct a real experiment to test their ideas about Alice The goal is to show that by using the U3 gate, instead and Bob [9]. We did find a real experiment, using mirrors, of one of the Pauli gate, a fifth state can be encoded and photon detectors, and other optic equipment, of sending a communicated by the single entangled qubit. The encoding single photon on a two-way journey from Alice to Bob and decoding operations will first be done using linear algebra. carrying two classical bits [10]. However, although real, they Once the mathematics have been demonstrated, the process did not use a real quantum computer. Best in our review was will be run on the IBM Quantum Experience quantum simula- cryptography research that used Alice and Bob, joined by Eve tor and finally on the IBM Quantum Experience functional the eavesdropper and Charlie the observer, in quantum key quantum computer. Each step will be explained in careful distribution that used quantum optics simulations that show detail so nothing is assumed other than linear algebra and basic real results that could be practical [11]. quantum code notation [6]. To run the experiment on the IBM Quantum Experience will also require familiarity with Qiskit We determined at that point that we must try to match Notebooks. quantum theory to its test in real quantum computer output with this simple example of Bob and Alice. We then quickly Section 2 gives a review of the literature on explaining experienced chaos as we attempted to execute that plan. It superdense coding using Bob and Alice. became obvious that this new ability to test linear equations of Section 3 details the requirements to duplicate the experi- gates on real quantum computers did not have firm sequential ment. steps for newcomers. Real quantum computer programming is raw, new, full of phenomena not well defined. We found Section 4 describes the process using linear algebra. in that chaos the second objective of this study: making our findings clear enough that they could be easy for our readers Section 5 explains the details of the quantum program to duplicate and follow in the real environment of quantum that was run on the IBM Quantum Experience simulator and computing. functional computer. Although some cryptographers dislike using the terms Section 6 presents the results. ”Alice” and ”Bob”, or, in fact, any human name, in quantum Section 7 gives the conclusion and describes possible future computing mathematics, and ask that researchers instead use work. symbols such as ”A” or ”B” [12], the 1978 seminal paper establishing RSA famously created Alice and Bob, and they II.LITERATURE REVIEW remain enduring tools to understand complex security communications protocols [13] including superdense coding, which is In the middle of 2017 came a great divide, as in that a purely quantum action - there is no classical equivalent. Alice time frame, IBM made available to the public real quantum and Bob are arguably as ubiquitous as Schrodinger’s cat [14]. computers over the Internet. IBM has continued this service, and the real quantum computers they are offering to the In 2018, Alice and Bob continue to appear in new quantum free service are in a state of continuously increasing qubit computer articles in publications from Russia [15] to China capacity. Research published before that time did not have the [16]. Alice and Bob are universal. We determined to use them, opportunity to test their equations on a commonly available as they are the standard. Previously Eve had been used as the real quantum computer. Research published after that time person creating the entangled qubits but this proved confusing could have included results from the nascent real quantum because in cryptography EVE is the eavesdropper, hence the computers. However, few do this. Alice and Bob and their change to Oscar as the originator. superdense example presents just the right simplicity for analysis on a real quantum computer. III.PROJECT REQUIREMENTS This study was done using the IBM Quantum Experience We did an extensive search of the literature on quantum and Qiskit Notebooks. Each individual wanting to duplicate computing since mid-2017 that expressed examples of Bob these experiments will need an API key and Qiskit which can and Alice. We found not a single paper presented evidence of be obtained at no cost from IBM’s Quantum Experience [17]. using a real quantum computer in their studies. The researchers The Qiskit Notebook implements the open source software de- mostly stayed in the familiar theoretical area of mathematical velopment kit (SDK), QISKit, that includes a Python API [18] proofs. We found intriguing research on quantum cryptography that translates the Python into Quantum Assembly Language where Alice locks her bit into a safe, and later gives Bob the (QASM) [19], which is then processed by an IBM quantum key, with the magic of superdense coding entanglement [7], computer [20]. a study of pure theory, no experiments. We also found where Alice and Bob each modify their quantum bit particles with Within the IBM Quantum Experience a new Qiskit Note- operators to reveal their superposition features and on the path book can be create by copying one of the existing Qiskit to each other, the bits bouncing, deflecting, off of interference tutorials. The Python quantum code included in section 5 from a unitary operator in their communication path that can be inserted into the new Notebook, replacing the existing influences their values - this is exceptionally interesting, yet the quantum code, to create a copy of the experiment. 2 The histograms used in this study are all direct output For lower left element of the matrix where φ = π/2, from The Qiskit Notebook and the IBM Quantum Experience √ √ iφ iπ/2 quantum computer [17]. e = e = eiπ = −1 = i . IV. METHODOLOGY For lower right element of the matrix where φ = π/2 and Superdense coding has been presented [21] [1] in a theo- λ = π/2, retical manner using a quantum diagram and notation of qubit ei(φ+λ) = eiπ = −1 values and has been demonstrated using Python programs that are written in Jupyter Notebooks and processed on the IBM . Quantum Experience quantum computer. [22]. The quantum circuit is represented by equation 2 The presentation is usually done using the the |φ+i Bell state, √1 (|00i + |11i) and Pauli gates. This allows superdense 2 coding to communicate two classical bits of information, H • U3 • H (2) four states, using only one of a pair of entangled qubits.The procedure is for the sender and receiver to agree on the meaning the four states, 00, 01, 10, and 11. The sender, who we have called Alice, decides what message she is going to First, the tensor product of the U3 gate and the identity get send and selects the appropriate Pauli gate or gates to apply to must be calculated to create a two qubit gate for the entangled her qubit. After encoding the data, she sends her qubit to the qubits. receiver, Bob. Bob applies a standard set of operations, CNOT with the Alice’s qubit as the control qubit and his qubit at   the target.This will disentangle the qubits. Then Bob applies 1 0 −i 0 1 1 −i 1 0 1 0 1 0 −i a Hadamard gate to Alice’s qubit. The results approach 100% √ ⊗ = √   probability for the value that Alice encoded. 2 i −1 0 1 2  i 0 −1 0  0 i 0 −1 The three Pauli gates are 180◦ rotations around their respective axis. The forth state is the original Bell state with no Applying the U3 gate to the |φ+i Bell state gives the gate, or the identity gate, I, applied. These gates give the nice following results. 100% results described above. The question I am addressing is what happens if you apply something other than a Pauli gate. For the research, the U3 rotation gate was chosen. This  1 0 −i 0   1   1  1 0 1 0 −i 1 0 1 −i gate has the matrix representation [23]. √   √   =   2  i 0 −1 0  2  0  2  i  0 i 0 −1 1 −1 cos (θ/2) −eiλ sin(θ/2) U(θ, φ, λ) = eiφ sin (θ/2) ei(φ+λ) cos(θ/2) Alice sends her qubit to Bob who proceeds to decode the message by first applying a CNOT gate. When we set θ, φ, and λ to π/2 the u3 matrix resolves to  1 0 0 0   1   1  0 1 0 0 1 −i 1 −i 1 1 −i     =   √  0 0 0 1   i   −1  2 i −1 2 2 0 0 1 0 −1 i The details of this calculation are as follows: The results are no longer entangled and can be factored The angles are measured in radians. There are 2π radians into a tensor product. in a circle, i.e. 360◦. From there we see that π radians = 180◦, π/2 radians = 90◦, and π/4 radians = 45◦. For θ = π/2  1  1 1 −i 1 1 1 cos(θ/2) = sin(θ/2) = cos(45◦) = sin(45◦) = 0.707 = √   = ⊗ 2 2  −1  2 −1 −i i This can be factored out of the matrix. By Euler’s identity, The right most factor is qubit 1, the high order qubit, whose value is eiπ + 1 = 0 or 1 iπ = |0i − |ii e = −1 −i . For the upper right element of the matrix where λ = π/2, √ √ The left most factor is qubit 0, the low order qubit. Apply −eiλ = −eiπ/2 = − eiπ = − −1 = −i the Hadamard gate gives. 3 VI.RESULTS 1 1 1 1 1 1 0 1 0 1 The histogram, figure 1, shows that the resulting counts √ = √ = √ = √ |1i 2 1 −1 2 −1 2 2 2 2 1 2 had a high probability of 01 and 11, table II.

State Probability When combined, the two qubits are 00 2.9% 01 47.1% 1 1 10 1.8% √ |01i − √ |i1i 11 48.2% 2 2 TABLE II: Distribution resulting from encode using u3 gate And when measured, the probability values are This is an entirely different result from that obtained by 1 1 (01), (11) encoding with one of the Pauli gates [6] and demonstrates the 2 2 ability to transmit a ﬁfth state using superdense coding.

V. QUANTUM PROGRAM Here is the code in Python used to create and entangle the two qubits, encode the message, and decode the message:

# Create 2 qubit quantum register q = QuantumRegister(2, ’q’)

# Create a quantum circuit circ = QuantumCircuit(q)

# Create entangle circ.h(q[0]) circ.cx(q[0],q[1])

# Alice’s encode circ.u3(pi/2,pi/2,pi/2,q[0])

# Bob’s decode circ.cx(q[0],q[1]) Fig. 1: Histogram of U3 encoding Superdense coding circ.h(q[0])

# Create 2 bit classical register c = ClassicalRegister(2, ’c’) VII.CONCLUSIONAND FUTURE WORK # Create measurement circuit meas = QuantumCircuit(q, c) It has been shown that running superdense coding in the meas. barrier(q) Alice and Bob scenario on a real quantum computer encoding with the U3 gate does transmit a unique ﬁfth state. # Map the quantum measurement to the classical bits meas.measure(q,c) Future work will focus on determining how many other, distinguishable, states can be achieved using other gates to # Combine quantum and measurement circuits encode the data. This includes how varying the angles used in qc = circ+meas the U3 gate affects results. # Specify the target IBM quantum computer stern = provider.get backend( ’ibmqx2’) REFERENCES

# Execute the circuit on the specified computer [1] M. A. Nielsen and I. Chuang, “Quantum computation and quantum # with 1024 iterations information,” 2002. j o b run = execute(qc, backend=stern , shots=1024) [2] A. Zeilinger, “Quantum teleportation, onwards and upwards,” Nature j o b monitor(job r u n ) Physics, vol. 14, no. 1, p. 3, 2018. [3] A. Ekert, “Quantum cryptography: The power of independence,” Nature # Extract the results Physics, vol. 14, no. 2, p. 114, 2018. r e s u l t r u n = j o b run.result() [4] P. Kaye, R. Laﬂamme, and M. Mosca, An introduction to quantum # Display the counts computing. Oxford University Press, 2007. counts = result r u n . g e t c o u n t s ( qc ) [5] E. Rieffel and W. Polak, “An introduction to quantum computing for non-physicists,” ACM Computing Surveys (CSUR), vol. 32, no. 3, # Generate the histogram pp. 300–335, 2000. from qiskit.visualization import p l o t h i s t o g r a m [6] L. Westfall and A. Leider, “Teaching quantum computing,” Future p l o t histogram(counts) Technologies Conference, 2018. in press. Listing 1: Superdense Coding Algorithm [7] M. Nagy and N. Nagy, “An information-theoretic perspective on the quantum bit commitment impossibility theorem,” Entropy, vol. 20, no. 3, p. 193, 2018.

4 [8] F. Del Santo and B. Dakic,´ “Two-way communication with a single quantum particle,” Physical review letters, vol. 120, no. 6, p. 060503, 2018. [9] P. Horodecki, M. Horodecki, and R. Horodecki, “Zero-knowledge convincing protocol on quantum bit is impossible,” Quantum, vol. 1, p. 41, 2017. [10] F. Massa, A. Moqanaki, F. Del Santo, B. Dakic, and P. Walther, “Ex- perimental two-way communication with one photon,” arXiv preprint arXiv:1802.05102, 2018. [11] G. K. Simon, B. K. Huff, W. M. Meier, L. O. Mailloux, and L. E. Harrell, “Quantiﬁcation of the impact of photon distinguishability on measurement-device-independent quantum key distribution,” Electron- ics, vol. 7, no. 4, p. 49, 2018. [12] R. Oppliger, “Disillusioning alice and bob,” IEEE Security & Privacy, no. 5, pp. 82–84, 2017. [13] R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Communications of the ACM, vol. 21, no. 2, pp. 120–126, 1978. [14] E. Schroedinger,¨ “The present situation in quantum mechanics,” Natur- wissenschaften, vol. 23, pp. 844–849, 1935. [15] S. A. Podoshvedov, “Quantum teleportation of unknown qubit beyond bell states formalism,” arXiv preprint arXiv:1801.09452, 2018. [16] K. Wang, X.-T. Yu, X.-F. Cai, and Z.-C. Zhang, “Probabilistic teleportation of arbitrary two-qubit quantum state via non-symmetric quantum channel,” Entropy, vol. 20, no. 4, p. 238, 2018. [17] IBM, Q Experience, 2020. https://quantum-computing.ibm.com/. [18] Python Software Foundation, 2018. https://www.python.org/. [19] A. W. Cross, L. S. Bishop, J. A. Smolin, and J. M. Gambetta, “Open quantum assembly language,” arXiv preprint arXiv:1707.03429, 2017. [20] Open Source Quantum Information Science Kit, 2020. https://qiskit.org/. [21] Michael Nielsen , Superdense coding: how to send two bits using one qubit, 2010. Available at https://youtu.be/w5rCn593Dig. [22] L. Westfall and A. Leider, “Superdense coding step by step,” in Future of Information and Communication Conference, pp. 357–372, Springer, 2019. [23] IBM, Qiskit-Terra, 2020. https://github.com/Qiskit/qiskit- terra/blob/master/qiskit/extensions/standard/u3.py.