IN
DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS
STOCKHOLM SWEDEN 2020,
Experimental Test of Local
Observer-Independence Using IBM’s Quantum Computer
LUDVIG EIDMANN MAX ALTEG
KTH ROYAL INSTITUTE OF TECHNOLOGY
Abstract
The precise interpretation of quantum mechanics is still open to discussion. A number of non-intercompatible interpretations, with a not unsignificant respective following, are at the time of writing still viable. When Eugene Wigner stated his titular thought experiment he managed to highlight how varying assumptions and accompanying mechanisms can have implications even on a macro-scale. Časlav Brukner expands the thought experiment of Wigner’s Friend in his recent paper and derives a theorem using the setup. In our paper, we empirically validate the theorem derived in Brukner’s paper, employing a setup based on his extended Wigner’s Friend experiment. More precisely, we realise this setup on a 5-qubit Quantum Computer and achieve a violation of a Bell-type inequality, close to what is theoretically possible. This seems to imply that local observer-independent facts are not possible within the current framework of quantum mechanics.
Keywords
Bachelor Thesis, Quantum Computing, Wigner’s Friend, IBM, Quantum Mechanics.
Sammanfattning
Den precisa tolkningen av kvantmekanik är ännu öppen för diskussion. Idag florerar det flera olika icke förenliga tolkningar, var en med ett inte insignifikant antal följare, vilka fortfarande är gångbara. När Eugene Wigner presenterade sitt självtitulerade tankeexperiment lyckades han lyfta fram hur olika antaganden och inbyggda mekanismer i dessa tolkningar kan ha implikationer på en makro-skala. I Časlav Brukners artikel expanderar han tankeexperimentet Wigners vän och härleder en sats med hjälp av denna uppställning. I vår uppsats validerar vi Brukners sats empiriskt med en uppställning baserad på det utvidgade Wigners vän experimentet. Mer specifikt utför vi experimentet på en 5-qubit kvantdator och uppnår en överträdelse av en Bell liknande olikhet där storleken på denna är nära vad som är teoretiskt möjligt. Detta tycks implicera att det inom ramverket för kvantmekaniken inte existerar några lokala och observatörsoberoende fakta.
Nyckelord
Kandidatexamensuppsats, Kvantdatorer, Wigners Vän, IBM, Kvantmekanik.
Contents
1 Introduction 1
1.1 Background . . . . 1
1.2 Problem . . . . 1
1.3 Purpose and Goal . . . . 2
2 Theoretical Background 3 2.1 Basic Quantum Theory . . . . 3
2.2 Bloch Sphere . . . . 4
2.3 Entanglement . . . . 6
2.4 The Measurement Problem . . . . 7
2.5 Wigner’s Friend . . . . 8
2.6 Bell’s Theorem and CHSH inequalities . . . . 8
2.7 A Theorem Concerning Wigner’s Friend . . . . 9
2.8 Qubits, Quantum Gates and Circuits . . . . 14
2.9 Architecture of Quantum Computers . . . . 15
2.10 Quantum State Tomography . . . . 16
2.11 Noise and Quantum Decoherence . . . . 17
3 Method 18 3.1 Construction of the Circuits . . . . 18
3.2 Optimizing the Circuits . . . . 21
3.3 Construction the Quantum State Tomography . . . . 21
4 Results 23 4.1 Obtaining a Violation . . . . 23
4.2 Error Analysis . . . . 23
5 Conclusion 27 5.1 Regarding Loopholes . . . . 27
5.2 Discussion . . . . 27
References 29
Appendices 29
1 Introduction
1.1 Background
Quantum mechanics was formulated in the beginning of the 1900s and was greatly expanded during the first half of the century. With the advancement of the theory came discussions regarding how to physically interpret what quantum theory seem to suggest is possible. One of the central topics of discussions was the implied phenomenon of entanglement, also called spooky action at a distance (section 2.3). The three prominent physicists Einstein, Podolsky and Rosen (EPR) jointly released their famous paper concerning this phenomenon in 1935 [8]. In this paper EPR argued for the existance of (now called) hidden variables which determined the behavior of quantum system but was not included in the theory. They hence argued that quantum theory was incomplete and needed to be expanded upon. This notion was later criticized by J. Bell who eliminated hidden variable theories under the assumption of what he called local realism [2]. To prove this he used what is now called the original Bell inequality. After the release of his paper, so called Bell tests sprung forth which could validate his thesis experimentally. Different versions of the Bell inequality exist where one of the most notable one is the Clauser–
Horne–Shimony–Holt (CHSH) inequality (section 2.6). The phenomenon of entanglement has been the foundation of several thought experiments where this phenomenon results in a contradiction when viewed from a classical framework. One of these thought experiments is the one stated by Eugene Wigner (section 2.5) which has managed to remain compelling even now.
1.2 Problem
Wigner’s thought experiment presents the discussion on what classifies as an observer, what classifies as a measurement and furthermore if and how the concept of consciousness has a place in quantum mechanics. Many versions of the thought experiments now exist and is still to this day an active field of research. A theorem has recently been derived [3]
based on one of these variations. It aims to concretize its implications, one of which is to eliminate the possibility of a hidden variable theory being viable. With the formulation of this theorem and the recent advancements in quantum computing it is now possible to realize Wigner’s Friend experiment on such a machine. The question is henceforth: will
the theorem be empirically validated?
1.3 Purpose and Goal
There are many fundamental questions without clear answers regarding quantum theory.
One method of examining these foundations is to pose thought experiments without clear answers, such as Wigner’s Friend. The goal of this thesis is hence to further the understanding of quantum theory via an experimental test of theorem 1 (section 2.7) using a quantum computer.
2 Theoretical Background
2.1 Basic Quantum Theory
A general quantum state vector |ψi is described as a sum over an orthonormal basis {ϕi}:
|ψi =∑
i
ci|ϕii, (1)
where ci are complex coefficents and the normalization requirement ∑
i|ci|2 = 1holds.
In vector notation this state is represented as
|ψi =
c1 c2
...
. (2)
A pure state is a quantum state represented by a single ket vector. A quantum state can also be represented by a density matrix ρ. For a pure state we have that ρ =|ψihψ|. In addition to pure states there are mixed states which are ensembles of pure states. Each pure state in the ensemble has an associated probability Pisuch that∑
iPi = 1. The density matrix for a mixed state is written as the sum of the respective density matrices of the included pure states, weighted by their associated Pi. A density matrix for a mixed state is therefore expressed as
ρ =∑
i
Pi|ψiihψi| [1]. (3)
The density matrix represents the probability for the possible outcomes of a measurement.
Physical properties which are measurable, such as momentum and spin, are called observables. An observable is manifested as a hermitian operator on the quantum state [15]. For an observable A with eigenstate |ϕai and an associated eigenvalue a we have
A|ϕai = a|ϕai. (4)
For every observable there exist a set of eigenvectors which make up orthonormal basis for the state-space and every eigenstate to an observable has an eigenvalue λi associated with it. A measurement an observable A on a quantum state|ψi will force the collapse of
the superposition to one of its eigenstates [15]:
A|ψi = λi|ϕii. (5)
Let A be an observable with only one non-zero eigenstate|ϕai:
A = λa|ϕaihϕa| = ρa. (6)
A measurement of A on the general state|ψi is described as
A|ψi = ρa|ψi =∑
i
λaci|ϕaihϕa|ϕii = λaca|ϕai. (7)
Equation (7) shows the measurement of A as a discontinuous projection of the original state along|ϕai. The probability of a measurement to yield a specific outcome λi, associated with the eigensate ϕi, is defined as
p(λi) = hψ|ϕiihϕi|ψi = |hϕi|ψi|2 [15]. (8)
The expectation value for an observable A is defined as the probabilistic expected value of the measurement and is defined as
hAi = hψ|A|ψi [15]. (9)
2.2 Bloch Sphere
A two level quantum mechanical system is a system which can only output one of the two possible values when measured. Such a systems is commonly called a qubit. For any two level quantum system there exist a geometrical representation called the Bloch sphere.
Every pure state|ψi of a two level system can be described on the form
|ψi = aeiχ|0i + beiξ|1i = eiχ(a|0i + bei(ξ−χ)|1i) = eiχ|ψ′i (10)
where a, b, χ and ξ are real. Due to the normalization requirement a2 + b2 = 1there is only one degree of freedom when assigning a and b. Let A be an observable acting on our
general two level system|ψi. We have that
hAi = hψ|A|ψi = e−iχhψ′|Aeiχ|ψ′i = hψ′|A|ψ′i. (11)
Hence only the phase difference ξ− χ can be physically observed and it is 2π periodic.
One can rename the available degrees of freedom; a = cos(θ2), b = sin(θ2)and ϕ = ξ−χ.
The Bloch Sphere can now be constructed with θ as the spherical polar angle and ϕ as the azimuthal angle. The axis of the sphere are defined as (figure 2.1):
ˆ
x : |+i = 1
√2(|0i + |1i), (12)
ˆ
y : |Ri = 1
√2(|0i + i|1i), (13)
ˆ
z : |0i. (14)
Each axis has a complementary orthogonal vector with which it is an orthonormal basis for the state-space:
|+i : |−i = 1
√2(|0i − |1i), (15)
|Ri : |Li = 1
√2(|0i − i|1i), (16)
|0i : |1i. (17)
Each of the three Pauli matrices (18) is diagonalized under one of the three basis.
σx =
0 1
−1 0
, σy =
0 −i i 0
, σz =
1 0 0 −1
. (18)
σx|+i = |+i, σx|−i = −|−i (19)
σy|Ri = |Ri, σy|Li = −|Li (20)
σz|0i = |0i, σz|1i = −|1i. (21)
Figure 2.1: Figure of the Bloch Sphere [11].
2.3 Entanglement
A two qubit system where one qubit is in the state|0i and one in the state |1i is described as
|0i ⊗ |1i =
1 0
⊗
0 1
=
0 1 0 0
=|01i. (22)
where the natural extended basis for a two qubit system is{|00i, |01i, |10i, |00i}. A pure state|ψi can be described as a product state, or a separable state, if there exist two states
|ψai and |ψbi such that
|ψi = |ψai ⊗ |ψbi. (23)
However, this does not hold for an entangled state which is therefore not separable [12].
A typical demonstration of this is to study the state (24) of two entangled particles.
√1
2(|01i + |10i. (24)
This state is not separable. A measurement of the first particle has a 50% chance for each outcome; |0i or |1i. Knowledge of the original state and the outcome of the first measurement grants information about a measurement on the second particle. This phenomenon of perfectly correlated outcomes of measurements is taken advantage of in a CHSH experiment (section 2.6).
2.4 The Measurement Problem
Quantum mechanics is formulated within a mathematical context where it is well understood and seemingly consistent. Complications seems to arise when applying quantum theory to describe the real world. An important example of this is known as the quantum measurement problem. This problem is often seen as multiple subproblems for which both the number and the exact wording varies [5][18][14]. One of these formulations describes the quantum measurement problem as one ”small” and one ”big”
problem [4]. The small measurement problem can be described as ”why a certain outcome - as opposed to its alternatives - occurs in a particular run of an experiment”. An answer to this question may or may not be trivial [4] but we will carry out no further investigation into this topic. The larger of the two problems has been formulated as ”what makes a measurement a measurement”.
There exist two distinct types of processes in quantum theory. First, the time continuous evolution of a quantum state can be described by applying unitary operators where the state of the system always exist in a deterministic sense. Second is the probabilistic and discontinuous projection also called the collapse of the quantum state. The big measurement problem seems to stem from the fact that, while in theory there exist a clear cut between these processes, this cut is not as clear in practice.
Consider a physicist who is to measure the polarization of a photon using a measurement apparatus. The physicist has no way of knowing the polarization before conducting the experiment and will hence describe it as a superposition of quantum eigenstates. Once the experiment is completed she will in her mind be sure of an answer to this question.
She must be equally sure that the superposition has collapsed at some point during the experiment. It may be tempting to say that the apparatus, when measuring the polarization, disturbed the photon and its quantum state. However, in a perfect experiment, the theory does not discriminate between the apparatus and photon as quantum systems. If one allows the apparatus to become entangled with the polarization of the photon one may place the cut between the physicist and her instruments. This would mean that when she views her indicator on the apparatus, she would collapse the quantum state. Once again, one will not find support in the theory that this is the necessary location for the cut and as we will see, this decision will have further consequences.
2.5 Wigner’s Friend
To shine a light on the real world consequences of the measurement problem Eugene Wigner created the self titled ”Wigner’s Friend” thought experiement [20]. Consider a similar setup as previously discussed with the polarization of a photon being measured by an observer, Wigner’s friend, in a lab. This perfect lab will in no way ”leak” information about the inside of the lab to the outside world, similarly to the box in Schrödingers cat.
The friend will measure the polarization in the orthogonal basis{|hi, |vi} as horizontal,
|hi, or vertical, |vi, where the state of the polarization before the measurement can be described as √1
2(|hi ± |vi). When the friend measures the photon he will from his point of view collapse the quantum state and simultaneously establish a ”fact”. Namely, that a definite outcome of the measurement of the polarization has occcured and that it has been recorded. Now consider a super-observer, Wigner, standing outside of the lab. Using quantum theory he will describe the combined state of the photon and the lab/record of the friend as a superposition:
√1
2(|”polarization is h”irecord⊗ |hiphoton±
|”polarization is v”irecord⊗ |viphoton).
(25)
Wigner can now in principle perform an interference experiment to confirm that the lab is in a superposition. Wigner establishes a ”fact” from his point of view, concluding that the friend cannot have recorded a definite outcome. This seems to indicate that ”facts of the world” depends on ones point of view. If the friend relays a message to the outside world stating ”I have recorded a definite outcome” (but not containing the result) the respective descriptions will still remain unchanged [19]. Will the friends measurement collapse the state available to Wigner and if not, are their worldviews reconcilable in any way?
2.6 Bell’s Theorem and CHSH inequalities
In response to the claims made by EPR on the incompleteness of quantum theory, Bell stated and proved his famous theorem in his paper from 1964 [2]. The theorem states that any local hidden variable theory cannot comply with all of the possible predictions of quantum mechanics [6]. This theorem was later expanded and can be put to the test experimentally in so called Bell tests. One of these Bell tests is the CHSH inequality
presented in the paper from 1969 [6]. The CHSH inequality can be expressed in an experimental context which makes it very useful for drawing mathematical conclusions from a real world experiment. A typical experiment is as follows: two entangled particles are to be measured by two different detectors: A and B. The outcome from a measurement of an entangled pair of particles can be either |00i, |01i, |10i or |11i. The detectors have different measurement settings: a, a′ and b, b′ for A and B respectively. With these measurements it is the possible to calculate the quantum correlation E(a, b) as
E(a, b) = N|00⟩− N|01⟩− N|10⟩+ N|11⟩
N|00⟩+ N|01⟩+ N|10⟩+ N|11⟩. (26) Here N|00⟩ denotes number of occurrences of measuring the entangled pair in the |00i state. The experiment proceeds to measure the quantum correlation for the four different settings. The CHSH inequality is then stated as
S = E(a, b)− E(a, b′) + E(a′, b) + E(a′, b′), (27)
|S| ≤ 2. (28)
A value of|S| greater than 2 is not permitted by any local hidden variable theory and would hence point to the invalidity of such a theory. It is however perfectly compatible with quantum theory, which predicts a maximum violation of|S| = 2√
2[16]. Entanglement inherently gives rise to correlations which are not possible using classical particles. A typical example of these correlations are those between particles separated at a large distances, where the influence of a measurement travels faster than light. The CHSH- experiment utilizes correlations that arise due to the outcomes of the measurement of the two particles being dependent. Measuring in a skew basis, one can reach a set of probabilities not attainable by classical systems. In other words, if the inequality is violated then that points to the measured system being a quantum system.
2.7 A Theorem Concerning Wigner’s Friend
Wigner’s Friend was recently addressed formally by Č. Brukner using an extension of the original setup [3]. The extended setup consists of two identical sets of laboratories each with a pair of super-observer and observer. A source will create a pair of entangled photons where one photon will be sent to each lab. Using this setup, Brukner is able to study and
draw conclusions in a mathematical context about Wigner’s thought experiment. Brukner has formulated the conclusions as a theorem.
Theorem 1. (No-go theorem for ”observer-independent facts”) The following statements are incompatible (i.e., lead to a contradiction)
1. ”Universal validity of quantum theory”. Quantum predictions hold at any scale, even if the measured system contains objects as large as an ”observer” (including her laboratory, memory etc.).
2. ”Locality”. The choice of the measurement settings of one observer has no influence on the outcomes of the other distant observer(s).
3. ”Freedom of choice”. The choice of measurement settings is statistically independent from the rest of the experiment.
4. ”Observer-independent facts”. One can jointly assign truth values to the propositions about observed outcomes (”facts”) of different observers.
Brukner points out: The word ”universal”, used in assumption 1, is done so in the sence of Peres [17]: “There is nothing in quantum theory making it applicable to three atoms and inapplicable to 1023 ... Even if quantum theory is universal, it is not closed. A distinction must be made between endophysical systems—those which are described by the theory—and exophysical ones, which lie outside the domain of the theory (for example, the telescopes and photographic plates used by astronomers for verifying the laws of celestial mechanics). While quantum theory can in principle describe anything, a quantum description cannot include everything. In every physical situation something must remain unanalyzed. This is not a flaw of quantum theory, but a logical necessity ...”.
This concept is well complimented by Bell’s notion of FAPP irreversibility [4]. A system is to be measured by an observer, using a measurement apparatus. ”If we assume ... that quantum mechanics is universally valid, then it is in principle possible to undo the entire measurement process. Imagine a superobserver who has full control over the degrees of freedom of the measuring apparatus. Such a superobserver would be able to decorrelate the apparatus from the measured system. In this process, the information about the measurement result would be erased. Seen from this perspective, “irreversibility” in the quantum measurement process merely stands for the fact that it is extremely difficult – but not impossible! – to reverse the process. It is irreversible for all practical purposes.”[4]
This leads to the conclusion that a universal quantum theory must also be irreversible merely FAPP. Combining the concepts of universality and FAPP irreversibility gives rise to three ideas. Firstly, that any object can be described by a subject (observer) using quantum theory, which itself can be described as an object by another subject. Secondly, that any subject described as an object must be the object for another subject. Thirdly, the implication of universality calls the whole notion of a measurement into question, something at the heart of the Wigner’s Friend thought experiment.
A final note: statements 2-4 can be replaced with Bell’s notion of local causality [16] and the theorem will still hold [3]. The current articulation of the theorem clarifies how each statement is relevant with regards to the proof of the theorem and enables further analysis of the implications of the theorem.
Proof. Figure 2.2 shows to sets of identical laboratories denoted as A and B. Under the assumption of universality both WignerA and WignerB can, similarly to the original version of Wigner’s Friend, simultaneously establish that a measurement of the respective friend has taken place and that the laboratory which the friend occupy exist in a superposition of states. A Bell test will be performed on this exterior quantum state available to the two superobservers. Each superobserver, WignerA and WignerB, will have two available measurement settings, A0, A1and B0, B1respectively. A superobserver can choose to either measure the state of the friends record (A0 and B0), verifying their measurement, or to perform a measurement along a different axis (A1and B1), establishing a fact of their own. For this proof we will define an observer as ”any physical system that can extract information from another system by means of some interaction, and store that information in a physical memory” [19]. This definition includes both systems as
”large” as human observers and as ”small” as a non-conscious quantum computer. The assumption of universality not only supports this definition but arguably implies it. Indeed, it is sufficient for observers acting in this proof to be able to perform measurements and record their outcomes.
Assumptions (2), (3) and (4), just as local causality, specifies the need for a local hidden variable theory. The phenomenon of entanglement inherently defies the notion of locality.
One can in response to this assert either the completeness of quantum theory and disregard locality all together, or that the theory is incomplete. A theory that includes the, not yet accounted by quantum theory, underlying mechanisms that governs the behaviour of
entanglement is what is now called a hidden variable theory. But as Norsen explains [16]:
”Bell’s historic contribution was a theorem establishing that no such local hidden variable theory—and hence no local theory of any kind—could generate the correct empirical predictions for a certain class of experiments.” This class includes the CHSH exeriments.
A local hidden variable theory that predefine the outcomes of the observables A0, A1, B0 and B1would imply the existence of a joint probability distribution p(A0, A1, B0, B1)[3].
The distribution must satisfy the CHSH inequality
|S| ≤ 2, (29)
where
S = hA0B0i − hA0B1i + hA1B0i + hA1B1i. (30) Here for example:
hA0B0i = ∑
A0,B0=−1,1
A0B0p(A0, B0) (31) and
p(A0, B0) = ∑
A1,B1=−1,1
p(A0, A1, B0, B1) (32) and similarly for other cases. We note that free will is used in the assumption that all quantum correlations will be evaluated. The proof of theorem 1 will, just as Brukners original proof, consist of a setup generating a quantum sate which is used to violate a CHSH inequality. A source will emit entangled qubits, a and b, in the state |ψ−iab =
√1
2(|0ia|1ib− |1ia|0ib). A rotation of angle θ around the ˆy-axis is then applied to qubit b, resulting in the state
|ψiab=1 ⊗ e−2iθσy|ψ−iab =−sinθ
2|ϕ+iab+ cosθ
2|ψ−iab, (33) where|ϕ+iab = √1
2(|0ia|0ib +|1ia|1ib). Two more qubits, α and β, are then entangled, representing the measurements of the friends inside the laboratories. The state after the entanglement is
| ˜Ψi = −sinθ
2|Φ+i + cosθ
2|Ψ−i, (34)
where:
|Φ±i = 1
√2(|00iaα|00ibβ± |11iaα|11ibβ), (35)
|Ψ±i = 1
√2(|00iaα|11ibβ± |11iaα|00ibβ). (36) In order to reach a maximum violation of the CHSH inequality; set θ = π4 (see Appendix A). We define two sets of binary observables as
Az =|00ih00| − |11ih11|, (37)
Ax =|00ih11| + |11ih00|, (38)
and similarly for B. The two observables play the same role as Pauli spin-operators along the ˆz and ˆx-axis respectively. Set Az = A0, Ax = A1, Bz = B0,and Bx = B1. As previously stated, the observable A0 represent a measurement of the friends record as it simply evaluates the state of two, in an ideal system perfectly correlated, qubits along the z-axis. Aˆ 1 can also be described as
A1 =|Φ+iaαhΦ+|aα− |Φ−iaαhΦ−|aα. (39)
This observable makes a measurement in the maximally entangled basis {|Φ+i, |Φ−i}.
Evaluating (30) with this choice of observables one reaches a value of S = −2√ 2. A violation of the CHSH inequality has been achieved:
|S| = 2√
2. (40)
|ψ−i
Observer entanglement Entangled state generator Rotation
a
b
a, α
b, β FriendA(α)
FriendB (β)
WignerA
WignerB
Figure 2.2: Diagram of the extended Wigner’s Friend experiment.
2.8 Qubits, Quantum Gates and Circuits
A classical computer uses a binary bit as a unit of information which is either on (1) or off (0). A quantum computer instead uses qubits. Each qubit is a two level quantum system which is operated on by quantum gates. These quantum gates are (sometimes closely) analogous to the logic gates used in classical computers. Quantum gates are unitary operators which can be represented by matrices using the canonical basis for one, two or more qubits. When a number of qubits are operated on by a set of quantum gates we call this a quantum circuit, often illustrated horizontal lines. Each line represents a qubit and is often ordered with qubit 1 (or 0) at the very top, followed by qubit 2 (1). Boxes are used to represent gates; seen in figure (2.3). The most general unitary operator on a single qubit is called the U3-gate which rotates a quantum state on the Bloch sphere.
U3(θ, ϕ, λ) =
cos(θ2) −eiλsin(θ2) eiϕsin(θ2) ei(λ+ϕ)cos(θ2)
(41)
Similarly, all single qubit gates can be visualized as a rotation around the Bloch sphere.
Each of the Pauli spin matrices σx, σy, σz can be visualized as a π-degree rotation around respective axis on the Bloch sphere. The Hadamard gate H which can be visualized as a combined rotation of π2 around both ˆxand ˆz-axis.
H = 1
√2
1 1 1 −1
(42)
Qubits can, simlilarly to all quantum systems, be entangled with one another. To do this one uses gates acting on more than one qubit. Analogous to the classical XOR logic gate is the Controlled Not gate (CN OT ), also known as the Toffoli gate. We will use the two qubit CN OT gate which applies an X-gate on the target qubit if the control qubit is|1i.
Since qubits can exist in a superposition of states this process can create a superposition of two qubit-states.
CN OT =
1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
(43)
The CN OT -gate is notated by a• on the control qubit and a⊕
on the target qubit, see figure 2.3. To make measurements in a quantum circuit one collapses the state of a qubit with the measurement gate. The gate project the state along the ˆz-axis and records the outcome as 0 or 1. In order to perform measurements along different axis one applies a unitary gate before the measurement gate. This is shown in the circuit of equation (44) where the H gate projects the state along the ˆx-axis before measurement. The state|+i would then be recorded as|0i on the ˆz-axis.
|0i H (44)
|0i X H •
|0i X
Figure 2.3: Example of a small quantum circuit.
2.9 Architecture of Quantum Computers
IBM has several quantum computers available to the public with differing number of qubits. Each quantum computer is designed using a specific architecture, a structure influencing how the qubits can interact with one another. To design an optimal quantum circuit with multiple cubit-gates (CNOT) one may need to take the architecture of the quantum computer into consideration. Only some pairs of qubits can be directly operated on by the CNOT gate. More specifically, referencing figure 2.4, only qubits at the tail of an arrow can act as the control qubit and one at the head of the same arrow as the target qubit [7]. More options are available through additional gates.
0
1
2
4
3
Figure 2.4: Architecture of IBM Vigo. Bidirected arrows indicate that the CNOT-gate can directly operate using either qubit as target.
2.10 Quantum State Tomography
In order to control that the state produced by the circuit is the one desired, one can perform a series of measurement on the state where it is projected on different basis. This is called Quantum State Tomography. With this method it is possible to reconstruct the density matrix of the state. Using the density matrix one can calculate the fidelity of the source which is a measurement of the quality of the produced state [1].
The density matrix for a single qubit state (3) can also be represented with Stokes parameters,{S0, S1, S2, S3} as
ρ = 1 2
∑3 i=0
Siσi. (45)
The Stokes parameters for a one qubit system are
S0 = P|0⟩+ P|1⟩, (46)
S1 = P|+⟩− P|−⟩, (47)
S2 = P|R⟩− P|L⟩, (48)
S3 = P|0⟩− P|1⟩, (49)
where each P|ψ⟩ correspond to the outcome of a specific projective measurement. More precisely; P|ψ⟩is the probability to measure the state|ψi [1]. This means that in order to reconstruct the density matrix for an unknown single qubit-state projections on each of the three axis of the Bloch sphere is necessary. For a two qubit system the density matrix is
constructed as
ρ = 1 22
i,j=3∑
i,j=0
Si,j σi ⊗ σj, Si,j = Si⊗ Sj. (50) Meaning that in order to make a complete state tomography of an unknown two qubits state it will require 16 measurements. However as S0,0 always will be equal to 1, only 15 measurements are needed in practice. The fidelity is calculated as
F (ρ, σ) =Tr(√√
ρσ√ ρ
)
, 0≤ F (ρ, σ) ≤ 1 [1]. (51)
Where ρ is the reconstructed density matrix and σ =|ψσihψσ| is the density matrix of the desired state, in this case|ψσi. If ρ = σ then the fidelity is equal to 1. If the two states ρ6= σ then the fidelity will be less than 1.
2.11 Noise and Quantum Decoherence
In a perfect quantum computer the produced quantum states would operate in total isolation. However this is not the case and the quantum state is subject to noise and quantum decoherence. Decoherence is seen as the loss of quantum information over time [9]. A classical interpretation of quantum decoherence is the loss of a systems energy due to friction. Every interaction between a quantum system and it’s environment results in a loss of quantum information. One example of this is gates operating on a quantum state.
To mitigate this effect the use of less gates and shallow circuits is preferred.
3 Method
3.1 Construction of the Circuits
As the system in 2.7 inherently uses qubits it can be implemented on a quantum computer without considerable alteration. Two qubits, a and b, are initiated in the state|0i. Qubit a is operated on by an X and an H-gate while qubit b is operated on by an X-gate. In order to entangle the qubits, a CN OT -gate is applied (see (52)-(55)).
X|0i = |1i (52)
H|1i = 1
√2(|0i − |1i) (53)
|ψ−i = CNOT ( 1
√2(|0i − |1i) ⊗ |1i )
= 1
√2(|01i − |10i) (54)
|0i X H •
|0i X
|ψ−i (55)
A rotation of the angle π4 around the ˆy-axis is applied to qubit b in order to achieve a maximum violation.
|0i X H •
|0i X U3(π4, 0, 0) |ψabi (56)
Each friend makes a measurement of its qubit and stores the result. Two additional qubits, denoted as α and β corresponding to a and b respectively, act as the friends’ memory,
”copying” the outcome of the measurement. Two CN OT -gates are applied, representing the measurements of the friends. Both pair of qubits, (a, α) and (b, β) are entangled, and hence all four qubits.
|0i
|0i X H • •
|0i X U3(π4, 0, 0) • | ˜Ψi
|0i
(57)
|ψiab|0iα|0iβ
CN OT⊗2
−−−−−→ | ˜Ψiaαbβ (58)
where| ˜Ψiaαbβ is the same as in (34). Our two observables (37)-(38) represent making measurements along the|00iaαand|Φ+iaα-axis respectively while assigning eigenvalues
±1 for each basis according to (59), similarly for (b, β).
Basis Eigenvalue
|00i +1
|11i -1
|Φ+i +1
|Φ−i -1
(59)
|Φ±i = 1
√2(|00i ± |11i) (60)
The observable A0 will be a measurement on the qubits a, α using the unaltered circuit and similarly for b, β. For the observable A1, a rotation around the ˆy-axis of π2 degrees will be applied before the entanglement of the auxiliary qubit α, again similarly for b, β.
An alternative but equivalent assignment of the two observables A1 and B1, which act only outside the quantum system that is to be measured, can be made. No alteration to the assignment of A0 and B0is needed since they act on the state| ˜Ψi directly. Using the fact that
CN OT2 =1 (61)
we can define the observable A1as the projection to the ˆz-axis after the operator
A1 = (CN OTaα⊗ 1bβ)(e−2iπ2σy⊗ 1α⊗ 1bβ)(CN OTaα⊗ 1bβ) (62)
has operated on the state| ˜Ψi. The equivalent operator for the observable B1is
B1 = (1aα⊗ CNOTbβ)(1aα⊗ e−i2π2σy ⊗ 1β)(1aα⊗ CNOTbβ). (63)
Combining all of the different measurement settings on both sides; A0, A1 and B0, B1, one can calculate a value for (30). Each observable hAiBji will assign eigenvalues to the respective eigenvectors and calculate the respective quantum correlation. Due to the design of the circuit all observables will assign +1 to the state vectors, |0000i, |1111i,
−1 to |0011i, |1100i, and zero to the remaining eigenvectors. In an ideal, noise-free environment no other outcomes than the ones associated with non-zero eigenvalues will
occur. This is however not the case for the system used in this experiment illustrated by the full dataset in Appendix D. All four combinations of observables are equally likely to be choosen under the assumption of free will. All circuits will hence run an equal number of times when evaluating equation (30).
|0i
|0i X H • •
|0i X U3(π4, 0, 0) •
|0i
Figure 3.1: Circuit used for measuringhA0B0i.
|0i
|0i X H • •
|0i X U3(π4, 0, 0) U3(π2, 0, 0) •
|0i
Figure 3.2: Circuit used for measuringhA0B1i.
|0i
|0i X H • U3(π2, 0, 0) •
|0i X U3(π4, 0, 0) •
|0i
Figure 3.3: Circuit used for measuringhA1B0i.
|0i
|0i X H • U3(π2, 0, 0) •
|0i X U3(π4, 0, 0) U3(π2, 0, 0) •
|0i
Figure 3.4: Circuit used for measuringhA1B1i.
3.2 Optimizing the Circuits
The experiment will run on the machine IBM Vigo, chosen due to its low error and simple architecture. Based on the connectivity scheme, to achieve the shortest possible depth of the circuit, the two lowest qubits b and β will be moved down one step. All CN OT -gates can with this modification be applied directly to the relevant pairs of qubits.
|0i
|0i X H • U3(π2, 0, 0) •
|0i
|0i X U3(π4, 0, 0) U3(π2, 0, 0) •
|0i
Figure 3.5: Optimized circuit used for measuringhA1B1i. All circuits used can be seen in Appendix C.
3.3 Construction the Quantum State Tomography
As previously stated in the theoretical background 15 different measurements are required to reconstruct the density matrix of the two qubit-source|ψ−i from equation (55). The 15 measurements alternate in projecting the qubits along different axis. When projecting the state along the ˆx-axis a H gate is implemented before the measurement. In order to project the state along the ˆy-axis a Rx gate is implemented which rotates the state around the ˆx- axis with an angle of θ = π/2. Projecting the first qubit along the ˆx-axis and the second along ˆz-axis generates the circuit seen in figure (3.6). Using this circuit it is possible to
