The interdisciplinary quantum information classroom:
Themes from a survey of quantum information science instructors

We developed an online survey drawing on prior experience with similar instructor surveys³³3Survey available upon request from lead author. [31]. We distributed the survey to faculty identified as potentially teaching or having taught courses containing significant QIS content. Emails were sent to 116 faculty with an invitation to participate in the survey in March-April 2021. Most recipients (109) were identified through the course catalog analysis discussed in Ref. [3], representing courses at 74 institutions. Additionally, faculty teaching courses that did not meet the strict definition of "introductory QIS course" given in Ref. [3] but whose courses were identified as potentially containing QIS content were also included in the initial email. We also sent the email to 7 faculty known to be QIS instructors who were not on the list produced by the search discussed in Ref. [3], and we asked faculty who received the email to forward it to any other faculty who might be interested in the project. A link to the survey was also distributed at a QIS education conference in June 2021 focusing on undergraduate education.

Faculty were asked to report basic demographic information about their course, such as the department(s) in which the course is listed, estimated cross-sections of the student population in the course by year and major, textbook used (if any), and any prerequisites or expected prior preparation in math, CS, or physics. We also asked faculty to identify specific student difficulties they observed and challenges they faced as instructors, as well as needs or requests for DBER work in the field of QIS. Questions were phrased as multiple-choice or free-response as appropriate. Prerequisite courses and student difficulties were qualitatively coded according to discipline and typical undergraduate course sequencing. A few longer free-response questions were used to elicit faculty’s broader opinions about issues pertinent to QIS education; transcripts from these questions were analyzed thematically.

We received 32 completed responses, 27 from the initial email link and 5 from the link distributed at the conference. We found it necessary to adopt a working definition of introductory QIS courses so as to screen courses with little QIS theory from the dataset, defining an introductory QIS course as a lecture course with a primary focus on theory or applications of QIS as defined by the four areas of fundamental science identified by the Subcommittee on Quantum Information Science [4]. Applying this definition, we discarded 4 responses from the dataset (2 courses focused primarily on materials science of quantum device fabrication and 2 traditional quantum mechanics courses with a brief QIS unit) giving a total of $N=28$ survey responses specific to introductory QIS courses.

One of the remaining 28 responses (Response #23) corresponded to an undergraduate course in development. We retained this response in our analysis of instructor and institution demographics and for aspects of the course that had already been established (e.g. prerequisites and cross-listing). We excluded it from research questions more directly focused on student or instructor experiences.

The survey was targeted at instructors of undergraduate and hybrid undergraduate-graduate courses, though we also received a total of 5 responses from instructors of exclusively graduate-level courses. We opted to include these 5 graduate courses in our analysis, given that a true introduction to QIS course will likely begin at a similar place academically whether the primary audience is at the undergraduate and graduate level. However, given the small number of graduate-only courses in our dataset and the undergraduate-oriented focus of our survey, our conclusions should be understood as applying most directly to undergraduate and hybrid-level courses. For this reason, none of the graduate courses were selected for follow-up interviews, and quotations from survey responses corresponding to graduate courses are explicitly identified as such.

A priority for our survey distribution was to ensure a diverse range of instructor backgrounds and institutions were represented. Institutions were classified according to the Carnegie Classification of Institutions of Higher Education [32] basic classification and institutional control (Table 1), as well as by geographic location. Findings are largely consistent with the distribution of courses discussed in Ref. [3]. As in Ref. [3], we find that private institutions and R1 research institutions are significantly overrepresented in our study compared to the cross-section of US institutions as a whole (Table 1). While we find that a strong majority of responses (68%) corresponded to doctoral-granting institutions, this rate is significantly lower than the 86% reported in Ref. [3], an effect we attribute to the survey distribution’s inherent bias in favor of undergraduate courses and the fact that beyond-intro QIS courses included in Ref. [3] excluded from the scope of this survey are likely to be disproportionately graduate courses. We also asked faculty to optionally report their race and gender; the respondents overwhelmingly self-identified as Caucasian and male (Table 2).

Institutions were also classified by MSI status using the November 2020 NASA List of Minority Serving Institutions [33]. Two institutions were identified as both AANAPISI (Asian American, Native American, and Pacific-Islander Serving Institutions) and HSI (Hispanic Serving Institutions). A third institution was identified as HSI but not AANAPISI. No other MSI categories were represented among responses.

We find that introductory QIS courses target students of diverse disciplinary cross-sections both within and across courses. We consider catalog listing and instructor estimates of the majors of students as indicators of the academic background for which the course is targeted. Given the diverse academic backgrounds of students, we find unsurprisingly that instructors teaching introduction to QIS courses perceive conflict between the goals of accessibility across disciplines and academic rigor; instructors take a variety of approaches in balancing these objectives, which we discuss below.

Figure 1 shows the academic departments in which each of the 28 intro to QIS courses is located or intended to be located. Where institutional catalogs distinguish between subfields of engineering, all listings classified as “Engineering” correspond either to Electrical Engineering (EE) or Electrical and Computer Engineering (ECE). We find that the majority of introductory QIS courses represented in this survey are listed in physics and/or computer science departments, though listings in EE/ECE and math are also represented in the sample. Fully 25% of the surveyed courses ($N=7$) are cross-listed in two or more departments, a strong indication that the instructor has an interdisciplinary audience in mind.

Given the uniquely interdisciplinary nature of QIS research, we anticipated that catalog listing may not reflect the actual diversity of academic backgrounds of students in introduction to QIS courses. Therefore, we also asked survey respondents to estimate the actual composition of their courses by major. Figure 2 shows instructors’ estimates of the distribution of majors for each of the 27 courses that had previously been offered. Enrollment profiles for each major are rounded to the nearest 5% corresponding to the typical granularity of estimates provided by instructors⁴⁴4We observed that all but one faculty member gave enrollment estimates as integer multiples of 5% or 10%. Though we have no independent means to corroborate the reliability of instructor estimates, we take faculty’s implicit consensus on the reliability of their own estimates as a reasonable bound on their overall reliability. Though we observe a rough correlation between catalog listing and actual course composition, the picture is more complicated at the level of individual courses. One of the cross-listed courses is 80% physics majors, for instance, while the majority of courses listed exclusively in physics or engineering physics (7/12) and computer science (5/7) had at least 20% majors from outside the listed department.

From the survey responses and follow-up interviews, we learned that many courses incorporate a diverse group of students by design. This intentional decision is illustrated by Carl, whose course features a near-even split between physics and CS majors. Though Carl’s background is in physics, he intentionally sought other perspectives from the beginning in recruiting co-instructors:

• Carl: We really wanted to work with someone in CS, too, because … we didn’t want there to be a CS version of the class, and a physics version of the class, and [an] ECE version of the class.

Carl’s course very intentionally appeals to a diverse audience of students. Whenever possible, each iteration of the course is taught by two or more faculty with different disciplinary and research backgrounds. The course is crosslisted in both the physics and CS departments. He is very explicit about his intent of designing the course for accessibility including across disciplines:

• Carl: Yes, our goal was [to] make this as accessible as we could.

Ben approached the design of his course with a similar intent of accessibility, having witnessed the same problem in a QIS course offered in a different department at his university:

• Ben: [The CS course’s] prereqs were pretty constrictive. You needed two or three intermediate-level computer science courses … so it wasn’t really open to non-CS majors … I tried to do the opposite – I tried to make it available to students in any department; they just needed linear algebra.

Unfortunately, perhaps because it was his first time offering the course, Ben’s efforts at accessibility across majors ultimately did not prove as successful. He had only one non-physics student, whose lack of quantum mechanics background appears to have caused some challenges:

• Ben: I had one student who was a CS major and had never taken modern physics. I’d hoped that the quantum computing course would be equally accessible to him … but this student had a little more trouble than the others.

Ben, Carl, David, and Edwin all aim to make their course widely accessible to a diversity of backgrounds, with varying motivations and degrees of success. This focus was not universal, however. Franz, whose course is in the CS department and caters primarily to CS majors, was explicit about his target audience in the interview:

• Franz: It’s not that I don’t like math, or physics – I love them! But this is a course for computer science students … so we’re very much … focused on algorithms … I’m focused like a laser on computer science.

Franz goes on to mention that he gets some math and physics students in his class – and even, if he recalls correctly, a linguistics student – but emphasizes that these students are not his primary focus, and he selects content accordingly. Franz’s experience appears to embody Seegerer, Michaeli, and Romeike’s observation that CS experts “agreed that a specific computer science perspective exists and is important” within QIS education [11].

Due to concerns about preparation, moreover, Albert indicated on his survey response that he had made explicit decisions to make the class more selective (though it was unclear how he accomplished this in practice):

• Albert: I am very selective about who can take the course, so I have very largely narrowed the possible gap between the Physics/Math major[s] and the CS majors.

And despite his focus on cross-disciplinary appeal, even Carl expressed the need to balance accessibility with rigor:

• Carl: I didn’t want to dumb it down … I want them to be able to actually do some of the real calculations, … see how Shor’s algorithm works … You know, there’s just a difference between Physics for Poets and Physics 1!

Edwin, in describing his accessibility-centered philosophy, cites a quote from Scott Aaronson that (perhaps inadvertently) contextualizes the trade-off we observe between accessibility and rigor: “Quantum mechanics is really easy if you take all the physics out of it.”

Traditionally, introductory QIS courses have been largely the purvey of graduate education. As discussed below, our findings suggest, consistent with Ref. [3], that QIS education has fully expanded to undergraduate education as well. While instructor interviews revealed tradeoffs due to differences in preparation, we heard from faculty that course design as well as the natural synergy among diverse groups of students make it possible to tailor the material for audiences of all levels.

Each course was classified by level (first-year [FY] undergraduate, beyond-first-year [BFY] undergraduate, or graduate) according to the course’s catalog number and the corresponding university definitions. Courses were classified as “hybrid” if they were listed for combined graduate and upper-division undergraduate audiences, either crosslisted across multiple course numbers or with a course catalog number explicitly indicating this status. Of the $N=28$ survey responses, 16 are listed (or are intended to be listed) at the undergraduate level: 2 dedicated first-year seminars and 14 BFY courses. An additional 7 courses can be classified as hybrid courses, while 5 courses are graduate-only. Interestingly, of the 3 courses listed or cross-listed in EE or ECE, all are graduate-exclusive courses despite our survey’s bias to the contrary, suggesting that the trend toward undergraduate QIS education may be occurring primarily in physics and CS.

Figure 3 shows the distribution of student years for each course, broken down by level classification. With the exception of the two first-year seminar courses, no course has greater than 10% freshmen, and most have similarly low levels of sophomores; the bias toward 3rd year and higher might be understood as a reflection of course prerequisites required by many introductory QIS courses as well as the audience toward which these courses are marketed. Also, Fig. 3 shows a close correlation between the actual course composition and the course level classification.

Similar to choices about which major or academic background(s) of students to target, decisions about what level(s) of students to target in a course also are subject to a series of tradeoffs. For example, one theme that emerged from a free-response survey question regarding student difficulties and gaps in understanding was that understanding quantum hardware and devices requires a depth accessible only to fourth-year undergraduates or graduate students, but that the class can be made more accessible for younger students by focusing on algorithms:

• Respondent #7: If one focuses on quantum algorithms, there is relatively little preparation needed … Any discussion on qubit implementations requires a lot of physics – quantum mechanics, condensed matter, etc. – and is only appropriate for seniors, if at all.

Respondent #7 was teaching at a primarily-undergraduate institution and thus made the decision to exclude most content on qubit implementations to match the student audience. Respondent #11 cited similar reasons for opting to teach a graduate-only course so that such material could be included. This challenge was particularly salient for FY seminar courses, as seen in Respondent #27’s survey response:

• Respondent #27: I do not assume any physics background, and this poses a limitation in explaining the hardware of various quantum computing platforms. Thus, I end up focusing primarily on theoretical aspects of quantum information, and giving only hand-waving explanations of the hardware involved …

Material that may be of interest to graduate students, particularly concerning hardware, can be inaccessible to undergraduates. One solution, adopted by Carl’s institution, is to offer separate undergraduate and graduate courses (Carl’s response focuses only on his undergraduate course):

• Carl: Well, one impetus for me [in developing the undergraduate course] was that occasionally undergraduates have taken … the graduate level class. And they do very poorly … it goes too fast.

Alternatively, we can look to hybrid undergraduate-graduate courses to see how instructors manage to bridge this gap. Ben designed a course targeting both undergraduate and graduate student levels by providing supplementary reading and homework for graduate students, particularly on physical realizations of quantum gates:

• Ben: For the undergrads … I think we may not have mentioned the Schrodinger equation! Whereas for the grad students, I had them read the chapters … [on] development of quantum gates from Hamiltonians.

David, on the other hand, argues that it is not particularly difficult to design a course meeting the needs of both undergraduate and graduate students. He reframes the diversity of student backgrounds – in discipline as well as level – as an instructional asset, with the diverse intellectual backgrounds of students combining to build a broad knowledge base in the fundamentals of quantum mechanics:

• David: It just – it didn’t seem unreasonable to be able to teach it at a level where a physics undergrad who … had taken the advanced quantum course or was concurrently taking the advanced quantum course. And … our graduate students, who had had more exposure. And in engineering who’d had more exposure to quantum mechanics already; they could all sort of reasonably take it at the same level, so – and it seemed to work.

In addition to the level and majors of students, we also asked survey respondents to report any prerequisites required for enrollment in their course. We find that the prerequisite courses in math, physics, and CS (if any) vary substantially across courses. We also find that course design is impacted by the inevitable trade-off between accessibility and the benefits of prior knowledge, with courses requiring fewer prerequisites having to cover more non-QIS background material.

Courses were classified as a prerequisite if they were required or strongly recommended for enrollment in the course, even if the instructor grants occasional case-by-case exemptions. Math methods courses in physics or CS courses were classified according to their relevant content areas.

Figure 4 shows the prerequisites given by the instructor for each of the 28 intro to QIS courses surveyed, broken out by topic (physics, CS, or mathematics). Prerequisites in other disciplines, such as engineering, were not asked about; among the courses that submitted optional syllabi, however, none listed prerequisite courses in such disciplines. Respondent #23’s course was included since the instructor had already determined the prerequisites he intended to require, even though the course had not actually been run yet. Note that most courses (19/28, 68%) require no CS prerequisite, and a similar fraction (18/28, 64%) require no physics prerequisite. Linear algebra is the most common prerequisite, required by 21 of 28 courses (75%), though many instructors allow students to select from multiple courses with linear algebra content.

One frequently reported challenge was the tension between the benefits and impracticality of requiring prerequisite courses. In the following quote, Carl discusses his teaching team’s decision not to require modern physics or discrete math as prerequisite courses, arguing that to require either would make the course inaccessible to a large portion of the intended audience for the course:

• Carl: When we were designing, we had a lot of discussion, like, it would really be better if we had [modern physics] as a prerequisite. But that limits us to physics students. It would be much easier if we had … the discrete math class the computer scientists take, but essentially no physics students take that.

Franz describes a similar dilemma in the context of his course targeting CS students at a small liberal arts college:

• Franz: Even though a course in quantum computing really could benefit from prerequisites in, like, classical circuit design, quantum theory, like more advanced linear algebra, more advanced like probabilistic algorithms, complexity theory, Fourier analysis – there are like all these things which the ideal student would have, but like, I don’t dare assume any of that, right?

Given the already small pool of students eligible to take the course, we asked Franz to elaborate on why unlike most instructors he required a second semester of CS coursework. His response was particularly informative:

• Franz: Actually, I doubt that we use really any material from the second course. So then why do I require it? … I felt like students who come out of our very first intro CS course, yes they’ve seen some computer science, but they’re still just not good at it, basically … It’s this kind of vague, I guess, argument of maturity.

The authors view Franz’s theme of maturity, echoed in similar language in the context of mathematics by Respondent #20, as suggesting that prior content exposure may be less important for students in intro QIS courses than mathematical and algorithmic sophistication.

Surprisingly, some instructors found that additional prerequisites, even when directly relevant to the content, are not actually helpful for student learning. Edwin has found no benefit in students completing a semester of undergraduate quantum mechanics, since the wave functions-first curriculum used at his institution does little to prepare students for the discrete vector spaces used in QIS. David related:

• David: We found that … how much quantum mechanics the students had previously been exposed to was not a very good indicator [of] how much trouble, or how well they did, in the course.

Respondent #20 even opted against a prerequisite in linear algebra – a prerequisite required by 75% of surveyed courses – having found, according to a statement in his syllabus, no statistically significant difference in student performance. In light of Respondent #20’s aforementioned emphasis on mathematical maturity – indeed, he offers optional out-of-class linear algebra review sessions to mitigate gaps in content coverage – we hypothesize that the disparity in instructor views on prerequisites reflects deeper disagreement on the role of prerequisite courses: are they prescribed to cover content or to build maturity?

In discussing the diversity of student backgrounds, particularly in the context of cross-disciplinary courses, a common theme emerged regarding the effects of diverse student backgrounds on the course as a whole. Some groups of students – usually physics or CS majors – would succeed more easily in one section of the course than in another. Additionally, student difficulties varied across majors. David reported that the diversity of students’ math backgrounds led to issues with boredom among students who already had strong math backgrounds. Likewise, some instructors discussed asymmetries between the backgrounds of physics and CS students, though this was not a universal experience. Carl reported that CS students tended to struggle with quantum measurement but had a much easier time with combinatorics than physics students. Edwin likewise noted that physics students tended to find the number theory and group theory associated with quantum algorithms particularly difficult. David reported that engineering students were much more likely than physics students to be comfortable with complex vectors. Regarding his graduate course, one survey respondent summarized:

• Respondent #19: Each major perform[s] better at different parts of the course.

Finally, multiple instructors reported needing to make adjustments to their teaching practices to accommodate the backgrounds of their students. Carl reported needing to make changes to the lecture slides due to incorrect assumptions about CS students’ math backgrounds. As we saw earlier, Ben found it useful to assign additional readings to graduate students to better align the level of the course to students’ background. Franz perhaps summarizes best the trade-offs associated with gaps in students’ backgrounds by noting that as a result, much of the material in the course won’t be directly about quantum computing:

• Franz: I have to explain how to factor integers, even though the jump from period finding [quantum subroutine in Shor’s algorithm] to factoring integers has nothing to do with quantum computing … Actually throughout the course, there’s a fair amount of material that’s just not quantum computing. And so that’s the kind of trade-off you make.

Consistent with their own diverse experiences and preparation, we observed a broad diversity of learning goals and intentions across the 6 faculty interviewees. We focus on faculty’s varied perspectives on the relative importance of two topics: fundamental physics and quantum programming.

Albert was emphatic that his primary goal was to build excitement and intuition for the underlying mathematical formalism and physics of quantum mechanics:

• Albert: I don’t give a damn about quantum computation, truth be told. I care about them learning quantum mechanics. That’s all I really care about.

Albert’s course accordingly eschews most quantum algorithms in favor of topics such as Bell’s inequality, and even focuses on the quantization of the electric field in order to physically realize a CNOT gate (a topic very much tangential to the scope of QIS as defined by Ref. [11]). David shares with Albert a strong focus on the underlying math and physics:

• David: The most basic thing is we’d like them to get – come away with a better understanding and ability to use the basic formalism of quantum mechanics, … the basic setup in complex vector spaces.

By contrast – and this connects with his choice to target computer science students as an audience – Franz is as adamant about the relative unimportance of fundamental physics to his course as Albert and David are to theirs:

• Franz: So I really … in some sense try to teach as little math and physics as possible.

Ben, Carl, David, and Franz all identified the ability to interpret and execute a quantum circuit diagram, computing the final state from the initial state, as an essential learning goal for their course. However, the natural extension of this idea – actually programming quantum computers – ranged in importance from essential to irrelevant in the views of respective interviewees. For example, Ben specifically wants his students to be able to translate between circuit diagrams and programming a real quantum computer on the cloud:

• Ben: Basically just looking at a quantum circuit diagram, understanding the Hadamards, the CNOTs, the other [quantum gates] – just understanding how to calculate what the final state is based on the [initial] state. And understanding how to run it on IBM Quantum on real remote devices and get data.

Edwin likewise considers the programming of a quantum computer an essential skill for the course. Franz doesn’t have his students extensively program quantum computers directly, but requires his students to code a classical simulation of a quantum computer. Meanwhile, Carl dispenses with programming entirely, arguing that it merely distracts from the fundamental focus on linear algebra and quantum gates:

• Carl: I think [programming is] very much beside the point of the topic of quantum computing. I think programming for quantum computers is 95% the same as programming for classical computers, and … teaching someone how to program classical computers just kind of isn’t necessary or of value when we’re trying to help people understand how quantum computers work.

Interestingly, none of the instructors have as a goal for students the ability to develop their own quantum algorithms. Franz explains that this goal would be impossible for his students because “they [quantum algorithms]’re too hard.” In his survey response, Franz elaborates:

• Franz: (A) Most of the intuition from classical CS doesn’t apply to the design of quantum algorithms, so we have to develop an entirely new intuition about how to design quantum algorithms, but (B) there are so few quantum algorithms (especially accessible, motivated ones) that we can’t really learn by example.

Given the interdisciplinary nature of QIS as a field, it is not surprising that some classes are co-taught by multiple instructors. David developed his course as part of a team of four, alongside one other physicist, one electrical engineer, and a former faculty member in electrical engineering and computer science whose research he describes as mostly mathematics. Additional input and guest lectures were provided by the wife of one of his collaborators, herself a professor in computer science and mathematics with a research focus on quantum measures of entropy. Likewise, Carl co-developed his course with two colleagues, one a fellow physicist and the other a computer science theorist whose graduate work involved quantum computing.

For David and Carl, having a diverse team was particularly useful given the ability to draw on the expertise of colleagues:

• David: [Two collaborators] were doing actual research in, in questions having to do with quantum information. So they were experts – I mean, not so much in the whole algorithm-type thing, but you know … certainly I learned an awful lot from them.

For David, who has relatively little background in QIS, working with a team has helped him solidify his own background. For Carl, himself an expert, having multiple disciplinary perspectives is a way to help students see the material from multiple angles. In both courses, there is the additional benefit that each instructor may teach the material that is most relevant to their own personal expertise; for instance, Carl’s lectures might focus on fundamental quantum mechanics while his computer science co-instructor would focus more on algorithms.

Relatively few of the instructors we surveyed and interviewed identified as experts in QIS. As a result, Albert, Ben, David, and Franz all expressed, in one context or another, the importance of proactive self-education in QIS education. Franz identified a particular challenge pertaining to self-education in the context of his background as a nonexpert. As a rapidly evolving discipline, teaching an up-to-date QIS course requires a knowledge of current research, yet primary literature published in journals is often not accessible to those without a background in QIS. As he writes:

• Franz: It’s not easy for me (as a non-specialist) to read the primary literature, and it’s really difficult to make it accessible to my undergraduate CS audience.

He suggests “digestible treatments of recent research developments” as a tool he as an educator would like, adding:

• Franz: It seems to me that QIS is young enough that a lot of people who are teaching it are non-specialists … I worry that my curriculum is 20 years out of date and I don’t even know it.

Franz’s comments suggest a broader need in the QIS community for more accessible scientific communication of novel results, aimed at a non-specialist but technical audience.

Survey respondents were asked to list student difficulties they observed in their course. We observed that student difficulties with mathematics were by far the most frequently reported category of student difficulty, being reported by 17 of 27 instructors who had previously taught at least one iteration of their course. Fully 14 instructors reported student difficulties with linear algebra. We find that bra-ket notation [34, 20], tensor products, and the spectral theorem⁵⁵5The spectral theorem, foundational to quantum measurement, states that a Hermitian matrix (the complex generalization of a real symmetric matrix) can be diagonalized in a basis of orthonormal eigenvectors with real eigenvalues. emerge as subtopics of linear algebra that are particular sites of instructor-reported difficulty.

Free-response survey answers were coded by topic, which were then grouped into broader emergent areas encompassing the full range of coded responses (mathematics, quantum mechanics, computer science/coding, other). We expect that this methodology will tend to underestimate the prevalence of specific student difficulties, since null responses do not necessarily imply the absence of a particular student difficulty.

Table 4 shows the student difficulties reported by instructors, as broken down by category. Notice that difficulties with mathematics dominate those of other categories, with 17 of 27 (63%) of surveyed instructors reporting one or more difficulties with mathematics. A smaller but significant fraction (37%) reported difficulties with quantum mechanics. Only one instructor noted a student difficulty with programming or related computer science topics, perhaps reflecting Carl’s sentiment that “physics students know more about computers than computer science students know about physics” and the observation that only a subset of instructors emphasize programming.

The right-hand column of Table 4 further breaks down the student difficulties in mathematics by subtopic. Note that 14 instructors cited one or more student difficulties with linear algebra, representing 52% of the survey respondents who had previously taught at least one iteration of their course and an overwhelming 82% majority of those respondents who reported at least one difficulty with mathematics. The large number of responses related to student difficulties with linear algebra is perhaps unsurprising, given the centrality of linear algebra in QIS. As Carl explains:

• Carl: Quantum computing to zeroth order is linear algebra.

Analysis of particular survey and interview responses provides greater insight into faculty perceptions of the nature of the student difficulties with linear algebra. Although students may enter the course able to perform matrix mechanics and manipulations, they lack the deeper understanding of, and intuition for, linear algebra necessary to apply the material to the abstract problems and vector spaces encountered in QIS:

• Franz: If we phrase linear algebra as like, can you multiply this $2^{N}$ matrix by this $2^{N}$ matrix … if we make it that concrete, I think my students have basically no problem with it – none. They start to get a problem when things are not written out that way, but are instead written out as like tensor products of states, right? Because that then is like a level of abstraction.

Franz further related that bra-ket (Dirac) notation was particularly challenging for his computer science students:

• Franz: I mean, arguably, it’s notation … there’s like kets flying around all over the place!

The student difficulties with bra-ket notation are consistent with Respondent #10’s observation that students came into the course lacking fluency in linear algebra even if they were familiar with basic concepts. Bra-ket notation abstracts away much of the concreteness of matrix notation in favor of the underlying, basis-independent mathematical structure. Students who are familiar with matrix mechanics but lack the fluency needed to abstract beyond rote calculations would likely struggle to learn bra-ket notation precisely for this reason. For instance, David finds that one particular student difficulty with bra-ket notation comes in forgetting that the symbol inside the bracket is merely an abstract label, giving the example that students will sometimes write $2\langle\psi|$ as $\langle 2\psi|$. David also finds that students struggle with applying gates to the correct qubit when translating between a quantum circuit diagram and bra-ket notation.

Both Carl and David emphasized tensor products as a particular difficulty for their students:

• David: A lot of our students end up at the core still confused about tensor products.

A major issue seems to be that linear algebra courses at the university level do not prepare students well for quantum computing; the topics and skills covered are simply too different. In fact, as Carl and Edwin point out, linear algebra concepts such as tensor product spaces and the spectral theorem, which are considered foundational to QIS, receive little to no attention in undergraduate linear algebra courses:

• Carl: So linear algebra classes, it turns out, do not teach you about tensor products usually, and even don’t talk about vector spaces much.

Of course, undergraduate introductory linear algebra courses are intended for applicability to a large range of majors; it is not realistic to expect these courses to equip incoming students with all the mathematical sophistication they require for success in a QIS course. Carl found that an intensive linear algebra review at the beginning of the course was effective in avoiding some of the pitfalls with student difficulties he experienced the first time he taught the course. And while general-audience linear algebra courses may never suit the needs of QIS students, discipline-specific math methods classes hold promise: Both David in physics and Carl’s co-instructor in CS have had success adapting their respective disciplines’ math methods classes to better emphasize the necessary mathematical skills and concepts for QIS.

Survey respondents were asked to report the textbook(s) required for their course, if any. There was relatively little consensus around any one particular textbook, and faculty perceived existing QIS textbooks as being poorly suited for undergraduate courses.

Of the 27 courses previously offered – Respondent #23 was still in the process of selecting a textbook at the time of the survey – a total of 14 unique textbooks (not counting separate editions of the same textbook) were reported as required by one or more courses. Of these, only 4 textbooks were required by more than one course, and only 2 textbooks were required by more than 2 courses: Quantum Computer Science: An Introduction, by N. David Mermin (2003), assigned by 7 faculty, and Quantum Computation and Quantum Information, by Michael A. Nielsen and Isaac L. Chuang (2011), assigned by 6 faculty. Tellingly, of the 27 courses, 9 courses assigned no textbook at all – in some cases drawing on extensive self-written course notes – while 6 courses required more than one textbook. David confirms that his primary motivation for writing his own course notes was textbook quality:

• David: All of this grew out of our dissatisfaction with the textbooks.

Complaints about the quality and suitability of existing textbooks were a prominent theme across survey responses, even though faculty were not specifically prompted to consider the quality of textbooks. A common theme was that existing textbooks are not appropriate for teaching upper-division undergraduate courses, either being “too mathematical” for undergraduates (in the words of Respondent #23) or not rigorous enough for the goals of the course:

• Respondent #16: All the texts I’ve looked at are either too low level or too high level.

Even among the faculty who utilized Mermin and/or Nielsen & Chuang, the consensus was that these textbooks were not optimal for undergraduates. In general, faculty perceived Mermin as too mathematically-dense and Nielsen & Chuang as requiring too much physics for undergraduates. As QIS education matures, the demand certainly exists for textbooks informed by education research and with an emphasis on current topics. However, the wide range of instructor goals suggests no textbook is likely to suit the needs of every course.