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Interviews with Farrukh Dhondy

The following transcriptions are of two conversations I had with Farrukh Dhondy, writer, left wing activist and former commissioning editor of Channel 4. 

We spoke about the origins of the poetry of Linton Kwesi Johnson, the fraught nature of the British Black Power movement in the late 1970′s and early 80′s.

These interviews were held on the 12th and 18th of November 2019. 

- Emily Blundell Owers

Tuesday 12th November 2019

 Farrukh Dhondy [FD]: (looking at printout of Voices of the Living and the Dead / VOTL&TD) I knew him in this phase, when he was formulating this. Well, you ask me the questions, I’ll tell you.

Emily [EB]: they’re not questions like ‘when did this happen’…I though that the first thing would be to tell you what the assignment is. ‘Take your text, and explain the relationship between what we have seen Raymond Williams call its social location and its aesthetic ideology. Each of these works is coming out of and speaking to a radical social movement at some moment in the twentieth century. I am asking you to research and reconstruct this context and to critically evaluate which elements of your work’s production and/or performance and/or publication/circulation history cast the best light on its actual or desired intervention in the world. Find out what you can about the decisions – personal, political – that inform the writing of your chosen work: its acts of aesthetic refusal and allegiance, its ways of thinking/doing community, uplift, justice, change. If appropriate, attend to any unintended effects or unconscious logics at play. Aim to address, in more or less equal measure, questions of literary form and of social world and to analyse both in relation to each other.’ Yes, it’s quite a long question.

FD: Good. Well it’s reassuring that you’re quoting Raymond Williams, who has gone out of fashion. But was very much in fashion when I was a student…before the postmodernists replaced him, who talk nonsense.

EB: So the first kind of questions that I was thinking were about the climate at the time…the 60s through to the 80s- this being first performed in 73 and published in 74. But my question is kind of…I get the impression that this was a time of social upheaval and what was it like in terms of one’s everyday experience?

FD: Let’s start there. I joined the Black Panther Movement (BPM/BP) in 1969. The reason was that I was a student from India and the entire movement consisted of immigrant workers, some immigrant students- Asians as well as West Indians- the movement was made of West Indian workers. And some people turned up in the youth branch of the BPM and they of course worked selling the newspaper, coming to meetings, coming to demonstrations that we organised, doing such things. Amongst them were the 6th formers of Tulse Hill school, one of which was Linton Kwesi Johnson. At the time he was not Kwesi, just LJ. But the BPM took its name through inspiration from the BPP of the United States. The BPP of the US departed from the civil rights movement of MLK, saying non-violence will NOT redeem the black population, or its social and political rights. We’ve got to take up arms, just as the American constitution guarantees that any citizen can carry arms and so on. And they were doing it as a piece of bravado- they weren’t actually going to shoot anybody- but they said we’ll carry guns and wear uniforms and this that and the other. But that was the inspiration, and the people who started the BPM here thought they’d take the name but it wasn’t the same struggle. We were immigrants from the ex-colonies, not descendants of slaves. Well yes, the West-Indians would come here, to work, who were- Indians weren’t. I was not- I came from a middle class family, on a scholarship to study at Cambridge, but I felt the same kind of tensions that Indian workers or west-Indian workers felt. Because there was a kind of race divide in Britain. So I joined that movement and the name of the movement was there to inspire black youngsters to join a movement which was dedicated to social and political rights in Britain. Get rid of racism. Get rid of police attacks on people. Get rid of pay differentials. Get rid of educational discrimination. Get rid of the fact that you couldn’t go into some places- a pub wouldn’t serve you, you couldn’t get housing- they’d say no blacks here. That had to be dealt with, it had to be dispensed with. One didn’t have to sit in different compartments, or at the back of the bus or some such, but there was a lot of discrimination going on in the 50s and 60s with the first wave of immigration, that’s the political background through which the BPM started, and which made people like Linton Johnson and others join up.

Now, there were radical teachers in Tulse Hill school. They were all white, but they were dedicated unionists, Trotskyists of sorts…they were the English department of THS. They were very active in the NUT, the national union of teachers, and of course they wanted an association with any other radical group. One got to know them. I was a teacher at the time.

EB: I heard that the Panthers spoke at the school, and that’s why Linton went along to the first meeting that he did…that it was through the school.

FD: Yes- they invited us to speak. I don’t know who spoke.

EB: I think it was Althea…

FD: Althea Jones. Yes. She was a postgraduate student in chemistry and biology, in London University. And she was very inspirational in as far as she was a good speaker. And these people invited her to speak there, and that’s how he got to know it and joined the BPM youth group, where we used to give history lessons in the Oval house, and we’d talk about ‘The Making of the English Working Classes’, Thompson, and other historical developments in America, in the West Indies, we wanted to know British history also. How it wasn’t exclusive- how the labour movement came about…it was all A.P Thompson’s book- it was very instrumental in those lessons. I had to give lectures on it, and young people were there. That was the political atmosphere in which that group began. And of course, in membership, you kind of devoted your life to it. Every day at some meeting, every weekend selling newspapers, during the week you were writing or having to publish it- Freedom News. And they’d call a movement (meeting?) on a Tuesday afternoon, and say ‘You’ve got to go somebody’s been arrested we are fighting for their release, outside the court or Brixton Police Station’. Constantly. So, we were doing all that. Agitational pamphleteering-

EB: Yes! Agit-prop.

FD: Exactly. And the group told itself that it aught to have its own educational internal systems. So, there were history groups, and some people wanted to start a literary movement. At that time, we had about 6 or 7 people who wanted to meet in a literature collective of the BPM, inside south London. And I was an aspirant writer myself- not aspirant, I was a published writer; I hadn’t written any books, but I’d written a lot of articles, lots of journalism and short stories- so they looked upon me as a kind of writer. And Linton wanted to be a kind of writer. And Linton turned up, we used to read things- if you sit in a collective of 8 or 10 people, in 33 Shakespeare road, in Brixton- you find out what other people are thinking, what they’re writing about. They’d bring their work and we’d criticise it, like a book club- a readers and writers association. At the time Linton was writing verse in imitation of what he had heard in 6th form English classes. He was writing not what he wrote later, but stuff that sounded to me like imitation T.S Eliot. Which he carried on into VOTL&TD. This book is not Jamaican patois.

EB: Yes. That’s a big thing, and its maybe more helpful to this question-

FD: Yes, this is (reads from text) ‘they came with fire blazing, death deep within our midst, desiring our destruction, we were water extinguishing their fire…’. Now this has a political bias. At the time, he was writing much more in the imitation of Eliot. Stuff he’d read and been inspired by his teachers to read. There was no notion in that literature group…that reggae was literature. There wasn’t. I think it was I who claimed that people are writing in the language that they grew up with- the language that they spoke at home, in the streets. And why…of course that was not something I invented. It was something I observed. There were poets in Jamaica and the West Indies who were trying to break out into dialect. There were other poets who wrote very purely in the English tradition. They were carrying on from Auden, Wordsworth, Keats, Eliot, Hughes. They were carrying on in that tradition even though they were black. But in the West Indies there was a movement to move towards the way that West Indians used English. Patois. Some call think that’s an insulting term.

EB: Yes, I’ve thought about this too- whether it’s creole or patois.

FD: Creole means ‘home language’.

EB: And then, coming off of that idea of the literary group, I’ve read Linton talking about reading in the Panthers, Fanon and Du Bois.

FD: Yes, in the history lessons…and CLR James of course. Whose biography I wrote. And he lived in our house, CLR James, when the kids were young. Yes, because his wife kicked him out. He stayed for a few months- he was supposed to stay 3 days! Anyway, so Johnson then went on with the inspiration of reggae and that debate which had been raging from the 30s in the west indies, over whether one should write in the British tradition, or apply bits of language and how people speak. And of course, he was influenced by that, and massively influenced by the reggae movement. And he began then- later on- VOTL&TD is kind of a construction in the British tradition, using the consciousness of hindsight. So, he has then put in the political content about tyranny, slavery…but the first branch out was when we told him, people told him, he became aware; ‘write in the language that your parents speak, how you speak on the streets, how Jamaicans speak’. And we got ‘Sonny’s Lettah’- I think that was probably the first branch out for him.

EB: I do find interesting, that 5NOB in this volume is also in DB&B. I was reading earlier, Brathwaite’s ‘History of the Voice’. And he analyses it, and writes it out in creole- in dialect. Because he’s analysing it from the EP, not how it was written down. And I thought that was interesting, because part of the question I’m asking myself is, because this is the reprint- rather than the original- why if this is in 83 that it was reprinted, so after DB&B has come out, after he started he publishing stuff in creole- I wonder why nothing was added to the volume that was this new type of poetry that he was writing.

FD: Braithwaite was at Pembroke college, Cambridge. Not at the same time as me, before. But I used to edit at Cambridge a magazine called Garconette (sp?), and I sought him out in London- said Mr Brathwaite, you were at my college. He probably thought ‘who’s this punk’- but I asked him to contribute a poem and he did. But it was not- he didn’t write- at the time in patois, he wrote in pure English.

EB: So, at the time he (Linton) was writing this (VOTL&TD), you don’t think he was experimenting with creole writing? Yet. Or maybe he was?

FD: Linton? I don’t know, but, as soon as Sonny’s Letter hit the decks, people though this was it, this is fantastic. And he never looked back. Except that when this was republished in 83- he went back to-he wasn’t ashamed of what he had written before. Did he revise it? Have you compared it?

EB: I can’t find 74 anywhere, but I’ve inferred from the introduction- which says ‘this volume’ was published in 74, and yes, I think it’s the same thing.

FD: You see, what would happen is that his teachers would have told him, and through Althea, write about the historical difficulties of what you think; Sonny’s Lettah is personal, he creates a character who is there. VOTL&TD is abstraction. He’s imagining that he’s the soul of black history, and that’s an abstraction. Putting himself in the place of the redemption of black history – ‘the tyrants came for us, they did this to us…’- they didn’t enslave him. He was walking around Brixton, having a drink. Smoking ganja. But he’s taking on the voice of the race.

EB: And calling for an uprising.

FD: Yes.

EB: I was going to ask about what Brixton was like in particular, and the relationships between minority youth, police-

FD: Brixton was very black- 100% black. There were no fashionable pubs, the Atlantic was a run down place, on the border of Coldharbour lane and Railton road- I lived at 74 Railton road. It was known as ‘the front line’.

EB: Yes, I’ve heard that Railton Road was a hotbed of different groups.

FD: What happened was, the BPM had a base in Shakespeare road- 38 SP road. That was given to us by somebody. Its first base was in Barnsbury, Islington. Very fashionable Islington. Because a rich white lady knew Althea and people, and said you can use my house to live in. You wouldn’t have known that it was the beginning of the BPM- it was in a very gentrified, even at that time- ‘Islingtonia’. And she handed over the house, so people would live in different rooms of the house. And the hall would be the meeting place for the Mangrove trials and so on. I would be up there every evening to write up what had gone on with the trial.

EB: I was reading about the Mangrove 9. And it was Franco Rosso, who did the film about the Mangrove 9…was it right that Race Today was based on Railton Road as well?

FD: What happened was, the Black Panthers then acquired property in Tollington Park. We had 3 branches in London. Shakespeare road, we used to call it the South London collective, then there was the West London collective, along the Mangrove- that gang, then there was the North London collective because they needed spaces across London, because there were members who used to come all the way South, as Althea said why don’t we base ourselves there. And as luck would have it, this writer, John Berger, gave us £2500, because he won the Booker Prize. And we, Darcus and I, with the BP, turned up- I was the only one with a fucking bank account. So we turned up to collect the money, with all these reporters trailing us and so on, and we took the money and put it into 37 Tollington Park, up near Finsbury park. And Althea and Eddie and everyone else moved in there. the entire story is told in a book I’ve written called ‘London Company’. Darcus was not a member of the central corps, because Althea didn’t like him.

EB: Of the Panthers? Why didn’t she like him?

FD: She said was a demagogue, a rabble rouser. And he was very attractive to the general membership. And he was a rival, so she kept him out of the central corps- saying he was a loose cannon, and he knew that. But one day they held a kangaroo trial…a chap called Brian, brought some white girl and the girls who lived there said we don’t want any white girls having sex in here. A household squabble. And they turned it into a kangaroo trial and called the central corps for an emergency meeting, saying ‘this man has disgraced the community’, and I said ‘What the fuck are you talking about? He’s a citizen he can fuck who he likes. It’s none of our business, we’re a political movement. He hasn’t broken any laws.’ And I said I’m not staying here for a kangaroo trial; I’m not doing this. And out of the 8 members of the central corps I walked out. And the first thing I did was phone Darcus, and said this is what’s happened. And he said ‘fuck, the whole place has deteriorated.’ So, we got a gang together, including Linton, and broke up the BPM, said fuck it. We want to do something else. So, we drifted for a while, didn’t go to the meetings, I signed over the house to Anthea and Eddie, her husband…but Linton came with us, he didn’t stay with the BPM, came with Darcus and me, and about 20 of us who quit the BPM started Race Today.

EB: That’s a really good story.

FD: So that’s how Race Today started and Linton joined it. And he continued his career as a poet, by this time of course he’d established himself by reading here and there. And the mood in Britain was towards celebrating Bob Marley, celebrating reggae, realising its rebellious- I wouldn’t say revolutionary- potential. And Linton tuned into that. It was very much attractive, to the establishment even. BBC 2 would do documentaries on him. That’s when Franco Rosso, everyone who thought they were on the radical left of the media, would join in to promote, celebrate, accentuate and bring to the public the ideas and voices they thought they were contributing to that rebellion or act of justice or revolution. All sorts of grades of ambition.

EB: So, at Race Today, were there specific roles?

FD: There weren’t specific roles. Darcus was certainly editor, he’d say what went into the magazine. The rest of us were writers, one of the first things I did for them was on the black explosion in British schools- a big article in the second edition.

EB: So, was Darcus Howe the editor from the start of the magazine?

FD: No, Darcus took over from a priest. Who used to run Race Today, when it was a magazine that belonged to the Institute of Race Relations in Kings Cross. And so, what happened was, Leila, who later on married Darcus, she and 2 or 3 others in the Race Today IRR said Alec was fine as editor, but we need a black editor. They were quite academic in their approach, the previous Race Today, just reporting this and that. They invited Darcus, who had just left the BPM, when we were floating about not belonging to anything, saying we’d make another movement and we’d have a black workers movement…they called him to be editor of Race Today, the collective of young black people who were working for IRR. And Darcus’ whole aim was, we’re not just going to be a magazine, we’re going to turn into an active collective.

EB: So the magazine was the start, and then it came out of that?

FD: Yes, and then he called me to write Black Explosion in British Schools, in February 73. The article is all about how the West Indian children of that generation that I was teaching are completely restless- by your generation they’d settled into the meritocracy, but in that generation they hadn’t.

EB: Are these second-generation kids, born here?

FD: Yes. Linton’s generation.

EB: This is something that I think is going to be the basis of my argument- the shift, the lack of complacency. So, I’m kind of wondering, for my own interest, your personal relationship with Linton. Because from what you’ve said it seems to be a kind of mentor relationship in the BPM.

FD: Hardly. He would never acknowledge anyone as his mentor. Of course, when he went his way and became a reggae poet, with groups giving him background to recordings and so on, there was no connection between me and him at all. We used to meet because we were part of the same collective, but otherwise there was no relationship with me whatsoever. I’ve never known whether he’s read my books- though I’ve followed his career.

EB: This was something I found interesting that I read today- in an interview he did with John La Rose, in I think around 97- he says, and I don’t think this will be a touchy subject from the conversation we’ve just had, but he said after he started touring the world with the successful albums, ‘I think there was a view within RT that I was too much of a high flier and my wings needed to be clipped’, in about 85. He said because of that he retired from music for 3 years, because he was needed for the Brixton organizational stuff.

FD: By 85 I had left Race Today, in the end of 83 I was appointed commissioning editor of channel 4, and I pulled in Darcus to do stuff. So, Linton was left with the collective and became one of its editors.

EB: I thought it was interesting.

FD: Yes, and maybe true. Its not that peoples wings…they probably said ‘what the hell are you doing? Do some political activity instead of cruising around!’

EB: I got the impression that it was more out of guilt than conscience, and I thought that was really interesting.

FD: You’re right I think, you’re right about that. And race today continued, Darcus left it because I gave him the money to fund Bandung files on Channel 4, where he started working full time. And Linton then began to run Race Today, and he must have felt responsible to do that. And if it was out of guilt rather than political conscience- well yeah, maybe. What he did do though, was bring in a lot of Jamaican poets- went to Jamaica, picked up Mikey Smith and others. And they all came and did a circuit of universities, and BBC2 came to the Race Today office and said they wanted to interview the poets.

 [recording cuts out here]

 Monday 18th November

 EB: So, I went through the things we’d spoken about before, to see what hadn’t been lost in the recording. I think the first question that cut off was regarding what it was like to be creative at a time when what was at stake was something more material…if the battles that you’re waging are more to do with police brutality and discrimination- threats which meant government at local and national level needed to be targeted, how can poetry get to that, or try to meet the requirements of an aim like that. I think we spoke about political consciousness raising?

FD: One can look at in several ways. A Caribbean population established itself here. It has its culture, its culture is religion, food- they may not be aware that all these things are known by the name of culture- what you wear, what you believe in, how you conduct the traditions within your life- all that becomes your culture. Now, art is a branch of culture. What you produce to be beautiful or instructive- not useful- a frying pan could be useful but that’s not art- but if you make a sculpture, a painting, if you write poetry or sing a song- those are creative things that come within culture, but they come within the subcategory of art. The workers who came here from the Caribbean had a culture for the Caribbean- the Trinidadians would sing calypso- there were black singers who had made it from the Caribbean- Harry Bellefonte. The people who came here didn’t quite discover what they were going to be writing about. They could do imitation calypso, they could imitate reggae and so forth, and some of them did try. But a particular culture began to evolve within British blacks, and Linton was certainly part of that. He knew that art went with particular forms, particular forms, particular moods and emotions. He had read English poetry, and American poetry, and knew that these were the things that poetry did. But, he had not discovered his subject, until he began to thing 2 things- 1) let me write in the language of the people that I live with, and am part of- actually youths. And the second thing was what should I write about. And it was always said, because we were in political turmoil, political action and struggle, combat and activation- it became clear to him that we always said write about what you are and what you do. It needn’t always be combative. It could be descriptive. When the philosopher C.L.R James said to us in a big meeting of the BPM ‘write about what you’re doing, and that will inspire other people rather than theories’. I was a schoolteacher at the time- I used to write 500 word articles for the paper about accusations of a black child stealing, or the 5th form disco, where fights went on, love rivalries- I wrote about that. James was telling us, write about what you know, about where you are. It will move other people to think in a similar way. Bus drivers wrote about how their union treated them, what happened at the garage, conductors about their interactions with the public- someone was racist to them, stuff like that. Linton would turn the experience that he saw around him into poetry, and that’s how he began writing Dread Beat & Blood, and then he took to writing actual propagandist poems- ‘Darcus Howe to Jail, Race Today cannot fail’, ‘free George Lindo’, stuff like that. So those were activist poems in support of a particular movement rather than descriptions of something that he had observed as an artist and poet. Those distinctions exist even within his work.

EB: Yeah, I think there’s even a further distinction in that this [VOTL&TD] comes before the absolute personal, there’s hints of it. He seems to be writing in the western tradition, of Eliot as you said. I found bits of ‘Voices’ very similar to ‘The Hollow Men’-

FD: Yes, that’s what he was doing.

EB: But he’s imparting upon that style-

FD: Yes- he discovered these two things. The language as his people speaks and the subject matter which he should now begin to represent. So, he did. And he was one of the first. Other people have followed, other writers. Now a whole spate of rap artists of your generation. Like Stormzy, someone who is a household name, who I know.

EB: Yes, my parents would know Stormzy, I doubt they’d know anyone else of that ilk. I suppose it’s similar to Linton, when we think about the documentaries- these people that want to be involved and push this figure.

FD: Sure, and of course what happens in our time is that art, even if it has propagandist and activist motive, becomes something to frame and look at. So even if you sing a song of love to woo a girl, it gets recorded and then its value doesn’t become the relationship, but how much the record will sell for. The function of art in our times has become that. Picasso paints ‘Guernica’, because he hates the Spanish civil war. What happens to the piece? Sells for millions. That’s the destiny of art in our times.

[break]

EB: Thinking about the performance of VOTL&TD at the Keskidee- I wish it were recorded somewhere but it’s not from what I can see. In some interviews he says there were dancers, in others dancers aren’t mentioned. And because obviously there’s different speaking voices- parts- I would really like to know if the audience were told before who each of them were, or if they were meant to infer it. Because if one voice is ‘the dead’ and one voice ‘the living’, I wonder how easy that is to recognise.

FD: Ask the George Padmore institute.

EB: Yes, I’ve looked at their archive website. But I’ll email. And I’ve got your book- I’m going to read the article on the black explosion in British schools, which from what I’ve glanced at is about the way black children were treated as inferior. Which I think is relevant, because I’ve read interviews with Linton that say in Jamaica he was top of the class, and when he came here they tried to put him in basically remedial lessons.

FD: That’s what happened in the first school I taught at. It’s not like that anymore- that’s gone. There’s still a lot of desperation amongst black boys though, gang crime.

EB: This question I think we spoke on, about the musicality of Linton’s work. About him being inspired by reggae artists- and I’ve also read him speaking about how he doesn’t consider himself to be a dub lyricist, because for him the words and the music were together as he wrote, rather than him needing a backing track for the poem to exist. Like reggae artists riffing on the track as it plays. I thought that was interesting because obviously 5NOB is the last in this collection, and I’ve seen it performed with music, but I thought it was interesting that it exists sans music, and I wonder if he wrote it for music, or if the music came after.

FD: What it is, when you look at his work- it’s extremely rhythmic. And it’s rhythmic in a drumbeat way. It has no complex rhythms which then come out, which some melodic songs do have- if you listen to Andrew Lloyd Webber, there’s different beats that progress within the songs. Linton’s poems don’t, they have the same – bum, pa dum, pa dum bu bu dum- and that actually accords well with the sort of instrumental music he did later. Because he recites, he doesn’t sing. Its not melodic, but it’s very rhythmic, and the rhythm is particular. And that’s just his style, the peculiarity of it. Lots of reggae songs, they use melodies found in popular music too- but his compositions as well are purely rhythmic too, the reggae beats.

EB: We spoke about Race Today, how the magazine was owned by IRR, and then when Darcus was appointed editor, he said he wanted it to be about activation, not just recording things. And I’d love for you to retell me that story about the post van-

FD: Yes. Darcus did a couple of months editing the magazine in the IRR at Kings Cross, and we determined in private talks that the magazine should not be – and so had they, those who worked there- had determined what they wanted to do. Because some of them were activists you see, in other groups like the Black Unity and Freedom party- who were doing things. They said to themselves, we don’t want the magazine to reflect the society, it’s not an academic magazine; we want to report the actions which we undertake, and we want it to stimulate the actions of people. If we wanted it to be like Lenin’s magazine in Russia, we needed to move it out of the academic atmosphere of the IRR. So one night a few of us, of course Darcus and I- with me driving my little old green post office van went to the IRR in Kings Cross, loaded up all the machines, the electric typewriters, pens, this that and the other – I don’t know if we took any desks, we may have done. The whole lot, we cleaned up the place. When they came in the next day they found everything gone.

EB: Was there bad blood?

FD: Yes. I think Darcus must’ve told Siavanandan, who thought he was the great leader of immigrants in Britain, who was a purely academic idiot. And not too academic- he didn’t know much. But he got himself into this position. He was a race peddler, not very bright- he’s dead now though, so its not libel. So we ran away, we established ourselves at 132 Railton Road, my second squat. 74 was my first squat, out of which I got burnt.

EB: How do you mean?

FD: I got burnt out. I was living on the second floor, 74 Railton Road, and at 4 o’clock in the morning someone through a bomb into the house. The house was ablaze, I woke up to the smoke. I thought someone was choking me, a pillow or something. I struggled, there was no pillow just burning smoke. I thought shit I’ve left the fire on, it’s caught fire. It wasn’t on.

EB: Was it people trying to get rid of the squatters or?

FD: No. It was the National Front. Because they hit 6 houses that night, all Asian and west-indian houses and shops. They fire-bombed. The fire engine police told me I was fire bombed. The police never caught them. It was on the 15th of march. 

EB: Beware the ides of march.

FD: Exactly. That’s when it happened. The 15th of March 1973. Burnt out of our house.

EB: It’s crazy to me that whenever I read about the New Cross house fire that it was never confirmed that it was the National Front. Who else would’ve done it?

FD: Electrical fault, somebody else, someone set fire to the basement because of a love rivalry. There’s all sorts of stories floating. Nobody knows I suppose.

EB: I can’t remember if I asked this last time- I wondered, not necessarily with this text as I don’t know how involved you were, but about the process of getting a text printed, getting somebody like Errol Lloyd to illustrate it. Do you think something like that would spring out of then already knowing each other, or-

FD: Well, the artistic world of blacks at that time was quite small, so they would have got to know each other, and they would have met at cultural occasions, where they were both speaking. And certainly the connection with Race Today could’ve helped- someone calling up and saying ‘I’m with Race Today, a poet, do you want to do my book?’

EB: I see, so it being a known thing people wanted to get involved. Also, not necessarily to do with Voices, but what do you think about capturing something in print which was made to be performed? Because on the first page of the poem, before it even starts, it says ‘with drums, bass guitar and flute’- and I suppose I’m asking for your opinion. Do you think illustrations in print are there in place of performance?

FD: No, they’re an additional form of art, apart from the performance. The performance is one thing, the illustrations their own form of art.

EB: Yes, this is why I’d love to see it. To see if there were costumes or-

FD: There wont have been costumes. I can tell you that. Youths of hope, for Darcus Howe…-

EB: Yes, that’s actually something I want to talk about, Darcus Howe. The fact it’s called ‘youths of hope’. I remember you saying to me before that Darcus was the driving force, was straight to the point.

FD: Yes, of Race Today. Yes.

EB: Everything I read about him, the mangrove 9, the black people’s day of action- every significant incident that is in the books I’ve read about any of this, Darcus is mentioned. So I guess the ‘youth of hope’ moniker is for him. It’s where I got a lot of my idea, my argument, about it being the younger generation who will be able to make a difference- rather than the complacency of the Windrush generation, it will be the younger people. I think 5NOB is my favourite one.

FD: It makes sense you see. Some of the others don’t- ‘terror tearing us up into pieces of smoke’- smoke doesn’t go have pieces!

EB: Well, I think YOH is helpful, when looking at VOTL&TD, to work out some things. Pinning down the tyrants, things like that. We spoke a little before on this- on whether at the time of writing this- because obviously all of this- even 5NOB which obviously made its way onto DB&B, is not written in the creole, the patois, any of the language that made him famous. Do you know if he was experimenting with it at that time- or do you think it was after? I remember you saying ‘sonny’s letter’ was the beginning, and after that he didn’t look back.

FD: I think so, I can’t be certain. 5NOB was refashioned- before he wasn’t doing anything in patois-

EB: Yes, it was actually really interesting to me that 5NOB, in his ‘History of the Voice’, Brathwaite analyses it, written out in patois- I think he did it from listening to the EP and writing it phonetically, because I’ve got DB&B, and it’s printed exactly the same as here. And that speaks to the fact that even though at this time Linton’s writing in standard English, if you were to hear him saying it, it becomes a completely different thing: his accent changes the whole thing. And it makes me think, how can a text that so needs to be heard in the voice of the person who made it- not what’s the point of printing it, because it’s obviously so more people can read it, but I find the tensions of that interesting.

FD: Yes. We all read Eliot for instance- and I’ve heard recordings of him reading Four Quartets, and it doesn’t give you a sense of a new dimension to the poem.

EB: So we don’t know for sure about the experimentations with creole, but I think I mentioned that this text we’ve been set is the republished, 2nd edition- so 9 years after the first in 1983. I wonder if you think there’s a particular reason for not expanding the volume, to include some of the work he was writing then. Because by then he’s published Sonny’s letter, and DB&B, released the albums which made him famous. He obviously wasn’t embarrassed of this stuff he’s done before.

FD: Well, even the publishers say ‘we don’t want collected works yet’, we’ll do them when you’re near to death.

EB: I’ve had to read a collected works of Linton!

FD: Well he’s getting old…

EB: My last question was- do you think Linton’s shift in writing style after this marked a shift in his intentions as poet. Maybe from politicisation to new means of expression for black creativity. VOTL&TD is riling up revolution of some kind and-

FD: Yeah, I think that the animus, the particularity of the activation has gone. There’s no Darcus in Jail or George Lindo- those were for particular publications of a monthly magazine- written for that. You couldn’t sell a poem in America or Jamaica, because nobody knows who that is, it’s a petty affair. Those poems were written for a particular audience, of a particular magazine, in a particular month. And they were just protest into rhyme. Of course he discovered that protest into rhyme works as journalism, but not as art. So I’m sure he’s shifted his focus to the attempt at permanence. The other thing, post-this, was that Linton befriended other poets from the Caribbean such as John La Rose. I don’t think La Rose is a good poet- or made an impact- but Linton befriended them, because Race Today deteriorated, got slower and went its way. Because frankly both Darcus and I left it.

EB: Do you think it was also kind of, not that the struggles had been won, but there was less-

FD: Times were changing certainly. And Linton then drifted off to the North to meet people like La Rose regularly, who of course he’d met in the Creation for Liberation times, World Book Fair and so forth.

EB: The way I’ve been thinking about it is like ’74, when this was written- there’s calls for uprising, calls for dying for what we believe in, even if that’s a lot of posturing. But I think even by the time this has been republished, there’s been the Brixton Riots, there’s been the battle of Lewisham. This stuff has happened.

FD: Yes.

EB: I think that you can attribute the change, that VOTL&TD is an attempt at politicisation, of whoever’s reading it, whoever’s watching; aimed at youth, the ‘youth’s of hope’, to stop ‘in-fighting’, start the righteous war- violence is justified if against police.

FD: Yes, he is saying that.

EB: And after this, it’s more like, ‘I’m speaking in my language…I’ve found my voice now’, like from the teaching in the BPM literary classes.

FD: Yes, adding the creole language to the literary tradition, maybe that’s what Linton will achieve.

EB: I think he knows that young people, 2nd generation children of immigrants, have got their own voice and that’s the protest and the riots and all this, is coming out of a lack of complacency. And the change, saying ‘I’m going to speak with this voice instead’, is a reflection of that.

FD: Yes.

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Pursuit of Excellence

Yasser Taqi, one of the two winners of NTIBizPro Awards 2019, is a paragon of grit and hard work, who does never make light of any task assigned to him. He is passionate about his job and fully dedicates himself to it, leaving no stone unturned in making it a success

Yasser believes that one of the hallmarks of an accomplished professional is the ability to align one’s skillsets with the changing dynamics of the market wherein one works. Therefore, he thinks, even extremely skilled professionals need to keep abreast of the changing trends in order to better position themselves in the market.

Impressive track record Testifying to this is his impressive education and professional track record. Yasser received, on completing his secondary school in Oman’s prestigious Sultan’s School, a scholarship from the Ministry of Higher Education to pursue a Bachelor’s Degree in Business with specialization in Business Law and Finance from Monash University in Australia. After his graduation, he returned to Oman to embark on his legal career as a paralegal for an international law firm in Muscat.

While working fulltime, he completed a Post Graduate Diploma in Law from BPP Law School in the United Kingdom and then completed a Postgraduate Diploma in Legal Practice (the Legal Practice Course also known as the LPC), which is prerequisite for practicing law in the UK. He was soon hired by Dentons, one of the world’s largest law firms, where he has been working for the last seven years.

Yasser has worked in three different offices with Dentons. He started in London, then moved to Dubai and eventually to Muscat. The experience has been quite enriching for him as it gave him a first-hand exposure of different ways of doing business and helped him build a personal network within one of the world’s largest law firms.

Yasser firmly believes in ‘work hard, play hard’ as way of life. “I give it my all when I am working; and I also try to enjoy and get the most of my time when I am not,” he says. This relentless pursuit of excellence coupled with an ability to take a broad spectrum view of the challenges he tackles, paid off well, when it came for NTI BizPro 2019. “Initially I was a bit intimidated, walking into a room with over 100 candidates taking an exam,” he recalls. “That exam can be daunting; but I told myself to take one step at a time; tackle each assessment as it comes and give it my best. It’s been quite an interesting experience. I had the pleasure to meet and interact with a number of very interesting people coming from various backgrounds.”

Broaden your horizon He says the youngsters in Oman are fortunate as they are blessed with all ingredients of success such as good education, quality infrastructure etc. But in order to utilise them efficiently, they have to work hard, identify the opportunities as they arise and broaden their horizons.

He asks the young Omanis aspiring to launch on their career not to shy away from work. “Roll your sleeves up and be ready to get your hands dirty, and always see things in the bigger picture,” he says. “Set yourself a goal, and see if what you are doing on a day-to day basis takes you closer to that goal.”

He is of the opinion that those who are starting off their career should get their priorities right. “They should not focus only on the monetary aspect of the job. At least to begin with, their focus should be on what job will give them better exposure and where do they learn the most, as opposed to where they earn the most. If you have the passion to do something and you do it well, money would follow,” he says.

He also encourages them to broaden their horizons, by keeping abreast of the wider global market and not limiting themselves to the local market. “In Oman, we have an incredible number of graduates who have passed out of reputed international universities, including the best universities in the world. And they do possess knowledge and amazing skillsets. But I think what we lack are people who go abroad and get the experience of working in major financial and industrial cities there and bring that experience back to Oman. And I think that is the way we have to take our nation to the next level,” adds Yasser.

Bouncing off ideas Yasser says he has been blessed to have several mentors wherever he has worked as well as in different walks of his life. “On a personal level, my family, parents and elder siblings have been immensely helpful. On a professional level, my elder siblings have always served as good role models for me. I have also been blessed with a number of mentors throughout my career who have helped me not only on the day-to-day works but also turning the spotlight on the big picture of my career trajectory. I am still in touch with most of them.”

He encourages young professionals to go and find their mentors. They have to bounce their ideas off with somebody with experience and insight. But he feels mentors do not have to be from the same industry. “One of my mentors with whom I closely work is based in London; he is a senior partner at the firm; the fact that he is in a different office, has a different focus and is in a very different stage in his career does not mean he cannot be my mentor. Mentors need not be your immediate bosses.”

Cooking and travel are some of Yasser’s favourite pastimes. He loves to meet new people, experience new culture and new ways of doing business. “I always like to try out new things, be it food, a destination or an activity,” he says.

Yasser, who was recently blessed with a baby girl, his second child, is looking forward to spending more quality time with his family. He aspires to be a leader not only in his professional domain, but also in giving back to the society by working with communities and helping youngsters achieve their dreams.

The post Pursuit of Excellence appeared first on Businessliveme.com.

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Towards Quantum PCP: A Proof of the NLETS Theorem

By Abhijit Mudigonda, Richard Wang, and Lisa Yang

This is part of a series of blog posts for CS 229r: Physics and Computation. In this post, we will talk about progress made towards resolving the quantum PCP conjecture. We’ll briefly talk about the progression from the quantum PCP conjecture to the NLTS conjecture to the NLETS theorem, and then settle on providing a proof of the NLETS theorem. This new proof, due to Nirkhe, Vazirani, and Yuen, makes it clear that the Hamiltonian family used to resolve the NLETS theorem cannot help us in resolving the NLTS conjecture.

Introduction

We are all too familiar with NP problems. Consider now an upgrade to NP problems, where an omniscient prover (we’ll call this prover Merlin) can send a polynomial-sized proof to a BPP (bounded-error probabilistic polynomial-time) verifier (and we’ll call this verifier Arthur). Now, we have more decision problems in another complexity class, MA (Merlin-Arthur). Consider again, the analogue in the quantum realm where now the prover sends over qubits instead and the verifier is in BQP (bounded-error quantum polynomial-time). And now we have QMA (quantum Merlin-Arthur).

We can show that there is a hierarchy to these classes, where NP MA QMA.

Our goal is to talk about progress towards a quantum PCP theorem (and since nobody has proved it in the positive or negative, we’ll refer to it as a quantum PCP conjecture for now), so it might be a good idea to first talk about the PCP theorem. Suppose we take a Boolean formula, and we want to verify that it is satisfiable. Then someone comes along and presents us with a certificate — in this case, a satisfying assignment — and we can check in polynomial time that either this is indeed a satisfying assignment to the formula (a correct certificate) or it is not (an incorrect certificate).

But this requires that we check the entire certificate that is presented to us. Now, in comes the PCP Theorem (for probabilistically checkable proofs), which tells us that a certificate can be presented to us such that we can read a constant number of bits from the certificate, and have two things guaranteed: one, if this certificate is correct, then we will never think that it is incorrect even if we are not reading the entire certificate, and two, if we are presented with an incorrect certificate, we will reject it with high probability [1].

In short, one formulation of the PCP theorem tells us that, puzzingly, we might not need to read the entirety of a proof in order to be convinced with high probability that it is a good proof or a bad proof. But a natural question arises, which is to ask: is there a quantum analogue of the PCP theorem?

Progress

The answer is, we’re still not sure. But to make progress towards resolving this question, we will present the work of Nirkhe, Vazirani, and Yuen in providing an alternate proof of an earlier result of Eldar and Harrow on the NLETS theorem.

Before we state the quantum PCP conjecture, it would be helpful to review information about local Hamiltonians and the -local Hamiltonian problem. A previous blog post by Ben Edelman covers these topics. Now, let’s state the quantum PCP conjecture:

(Quantum PCP Conjecture): It is QMA-hard to decide whether a given local Hamiltonian (where each ) has ground state energy at most or at least when for some universal constant .

Recall that MAX--SAT being NP-hard corresponds to the -local Hamiltonian problem being QMA-hard when . (We can refer to Theorem 4.1 in these scribed notes of Ryan O’Donnell’s lecture, and more specifically to Kempe-Kitaev-Regev’s original paper for proof of this fact.) The quantum PCP conjecture asks if this is still the case when the gap is .

Going back to the PCP theorem, an implication of the PCP theorem is that it is NP-hard to approximate certain problems to within some factor. Just like its classical analogue, the qPCP conjecture can be seen as stating that it is QMA-hard to approximate the ground state energy to a factor better than .

Reformulation: NLTS conjecture

Let’s make the observation that, taking to be the ground state energy, the qPCP conjecture sort of says that there exists a family of Hamiltonians for which there is no trivial state (a state generated by a low depth circuit) such that the energy is at most above the ground state energy.

Freedman and Hastings came up with an easier goal called the No Low-Energy Trivial States conjecture, or NLTS conjecture. We expect that ground states of local Hamiltonians are sufficiently hard to describe (if NP QMA). So low-energy states might not be generated by a quantum circuit of constant depth. More formally:

(NLTS Conjecture): There exists a universal constant and a family of local Hamiltonians where acts on particles and consists of local terms, s.t. any family of states satisfying requires circuit depth that grows faster than any constant.

To reiterate, if we did have such a family of NLTS Hamiltonians, then it we wouldn’t be able to give “easy proofs” for the minimal energy of a Hamiltonian, because we couldn’t just give a small circuit which produced a low energy state.

Progress: NLETS theorem

-error states are states that differ from the ground state in at most qubits. Now, consider -error states (which “agree” with the ground state on most qubits). Then for bounded-degree local Hamiltonians (analogously in the classical case, those where each variable participates in a bounded number of clauses), these states are also low energy. So any theorem which applies to low energy states (such as the NLTS conjecture), should also apply to states with -error (as in the NLETS theorem).

To define low-error states more formally:

Definition 2.1 (-error states): Let (the space of positive semidefinite operators of trace norm equal to 1 on ). Let be a local Hamiltonian acting on . Then:

is an -error state of if of size at most s.t. .

is an -error state for if s.t. and is an -error state for .

Here, see that is just the partial trace on some subset of integers , like we’re tracing out or “disregarding” some subset of qubits.

In 2017, Eldar and Harrow showed the following result which is the NLETS theorem.

Theorem 1 (NLETS Theorem): There exists a family of 16-local Hamiltonians s.t. any family of -error states for requires circuit depth where .

In the next two sections, we will provide background for an alternate proof of the NLETS theorem due to Nirkhe, Vazirani, and Yuen. After this, we will explain why the proof of NLETS cannot be used to prove NLTS, since the local Hamiltonian family we construct for NLETS can be linearized. Nirkhe, Vazirani, and Yuen’s proof of NLETS makes use of the Feynman-Kitaev clock Hamiltonian corresponding to the circuit generating the cat state (Eldar and Harrow make use of the Tillich-Zemor hypergraph product construction; refer to section 8 of their paper). What is this circuit? It is this one:

Image from [2]

First, we apply the Hadamard gate (drawn as ) which maps the first qubit . Then we can think of the CNOT gates (drawn as ) as propagating whatever happens to the first qubit to the rest of the qubits. If we had the first qubit mapping to 0, then the rest of the qubits map to 0, and likewise for 1. This generates the cat state , which is highly entangled.

Why do we want a highly entangled state? Roughly our intuition for using the cat state is this: if the ground state of a Hamiltonian is highly entangled, then any quantum circuit which generates it has non-trivial depth. So if our goal is to show the existence of local Hamiltonians which have low energy or low error states that need deep circuits to generate, it makes sense to use a highly entangled state like the cat state.

Quantum circuits

Image from [2]

(We’ll write that the state of a qudit – a generalization of a qubit to more than two dimensions, and in this case dimensions – is a vector in . In our diagram above, we’ll see 4 qudits, labelled appropriately.)

Let’s briefly cover the definitions for the quantum circuits we’ll be using.

Let be a unitary operator acting on a system of qudits (in other words, acting on ), where . Here, each is a unitary operator (a gate) acting on at most two qudits, and is a product of such operators.

If there exists a partition into products of non-overlapping two-qudit unitaries (we call these layers and denote them as , where each here is in layer ) such that then we say has layers.

In other words, has size and circuit depth .

Lightcones, effect zones, shadow zones

Consider and an operator.

For define as the gates in layer whose supports overlap that of any gate in , …, or with .

Definition 3.1 (lightcone): The lightcone of with respect to is the union of : .

So we can think of the lightcone as the set of gates spreading out of all the way to the first layer of the circuit. In our diagram, the lightcone of is the dash-dotted region. We have , , and .

We also want a definition for what comes back from the lightcone: the set of gates from the first layer (the widest part of the cone) back to the last layer.

Define . For , let be the set of gates whose supports overlap with any gate in .

Definition 3.2 (effect zone): The effect zone of with respect to is the union .

In our diagram, see that , , and . The effect zone of is the dotted region.

Definition 3.3 (shadow of the effect zone): The shadow of the effect zone of with respect to is the set of qudits acted on by the gates in the effect zone.

In our diagram, the first three qudits are effected by gates in the effect zone. So .

Given all of these definitions, we make the following claim which will be important later, in a proof of a generalization of NLETS.

Claim 3.1 (Disjoint lightcones): Let be a circuit and operators. If the qudits acts on are disjoint from , then the lightcones of and in are disjoint.

Toward the Feynman-Kitaev clock

Now we’ll give some definitions that will become necessary when we make use of the Feynman-Kitaev Hamiltonian in our later proofs.

Let’s define a unary clock. It will basically help us determine whatever happened at any time little along the total time big . Let . For our purposes today, we won’t worry about higher dimensional clocks. So we’ll write , but we’ll really only consider the case where , which corresponds to . For simplicity’s sake, we will henceforth just write .

Our goal is to construct something a little similar to the tableaux in the Cook-Levin theorem, so we also want to define a history state:

Definition 4.1 (History state): Let be a quantum circuit that acts on a witness register and an ancilla register. Let denote the sequence of two-local gates in . Then for all , a state is a -dimensional history state of if:

where we have the clock state to keep track of time and is some state such that and . With this construction, we should be able to make a measurement to get back the state at time .

Proof of NLETS

We provide a proof of (a simplified case of) the NLETS theorem proved by Nirkhe, Vazirani, and Yuen in [2].

Theorem 2 (NLETS): There exists a family of -local Hamiltonians on a line (Each Hamiltonian can be defined on particles arranged on a line such that each local Hamiltonian acts on a particle and its two neighbors) such that for all , the circuit depth of any -error ground state for is at least logarithmic in .

First, we’ll show the circuit lower bound. Then we’ll explain why these Hamiltonians can act on particles on a line and what this implies about the potential of these techniques for proving NLTS.

Proof: We will use the Feynman-Kitaev clock construction to construct a -local Hamiltonian for the circuit : .

Fix and let have size . The Hamiltonian acts on qubits and consists of several local terms depending on :

We can think of a qubit state as representing a step computation on qubits (i.e. for each time , we have a bit computation state of ). Intuitively, a qubit state has energy with respect to iff it is the history state of . This is because checks that at time , consists of the input to . Each checks that proceed correctly from (i.e. that the th gate of is applied correctly). Then checks that at time , the output is . Finally, checks that the qubit state is a superposition only over states where the first qubits represent “correct times” (i.e. a unary clock state where time is represented by zeros followed by ones).

Therefore, has a unique ground state, the history state of , with energy :

Later we will show how to transform into a Hamiltonian on qutrits on a line. Intuitively, the structure of allows us to fuse the time qubits and state qubits and represent unused state qubits by . For the Hamiltonian , the ground state becomes

For the rest of this proof, we work with respect to .

Let be an -error state and let be the subset of qutrits such that . We define two projection operators which, when applied to alone, produce nontrivial measurements, but when applied to together, produce trivial measurements.

Definition 5.1: For any , the projection operator

projects onto the subspace spanned by on the th qutrit.

For any , the projection operator

projects onto the subspace spanned by on the th qutrit.

Claim 5.1: For , . For , . Note that these values are positive for any .

Proof: If , then measurements on the th qutrit are the same for and .

If , then any qutrit pure state cannot have nonzero weight in both and (every pure state ends in some number of s which tells which (if any) it can be a part of). Therefore,

If , then projecting onto the th qutrit gives with probability . Therefore, .

Similarly, .

Claim 5.2: For such that , .

Proof: As before, we can calculate

If , then the th qutrit of is so . If , then because the first qutrits of contain the state so under any measurement, the and th qutrits must be the same.

Now we use these claims to prove a circuit lower bound. Let be a circuit generating (a state with density matrix) . Let be the depth of .

Consider some . For any operator acting on the th qutrit, its lightcone consists of at most gates so its effect zone consists of at most gates which act on at most qudits (called the shadow of the effect zone).

Assume towards contradiction that . Then the shadow of any operator acting only on the th qutrit has size at most since . So there is some outside of the shadow which is in the complement of . By Claim 3.1, we have found two indices such that any pair of operators acting on and have disjoint lightcones in . WLOG let . The lightcones of are disjoint which implies

By the two claims above, we get a contradiction.

Therefore, . We can take any constant epsilon: letting , we get

This analysis relies crucially on the fact that any -error state matches the groundstate on most qudits. However, NLTS is concerned with states which may differ from the groundstate on many qudits, as long as they have low energy.

Remark 2.1: The paper of Nirkhe, Vazirani, and Yuen [2] actually proves more:

A more general lower bound: logarithmic lower bound on the circuit depth of any -approximate ( far in L1 norm) -noisy state (probability distribution over -error states).

Assuming QCMA QMA (QCMA takes a bit witness string instead of a qubit state as witness), they show a superpolynomial lower bound (on the circuit depth of any -approximate -noisy state).

“Approximate qLWC codes”, using techniques from their superpolynomial lower bound.

Back to NLTS – Tempering our Optimism

So far, we’ve shown a local Hamiltonian family for which all low-error (in “Hamming distance”) states require logarithmic quantum circuit depth to compute, thus resolving the NLETS conjecture. Now, let’s try to tie this back into the NLTS conjecture. Since it’s been a while, let’s recall the statement of the conjecture:

Conjecture (NLTS): There exists a universal constant and a family of local Hamiltonians where acts on particles and consists of local terms, s.t. any family of states satisfying requires circuit depth that grows faster than any constant.

In order to resolve the NLTS conjecture, it thus suffices to exhibit a local Hamiltonian family for which all low-energy states require logarithmic quantum circuit depth to compute. We might wonder if the local Hamiltonian family we used to resolve NLETS, which has “hard ground states”, might also have hard low-energy states. Unfortunately, as we shall show, this cannot be the case. We will start by showing that Hamiltonian families that lie on constant-dimensional lattices (in a sense that we will make precise momentarily) cannot possibly be used to resolve NLTS, and then show that the Hamiltonian family we used to prove NLTS can be linearized (made to lie on a one-dimensional lattice!).

The Woes of Constant-Dimensional Lattices

Definition 6.1: A local Hamiltonian acting on qubits is said to lie on a graph if there is an injection of qubits into vertices of the graph such that the set of qubits in any interaction term correspond to a connected component in the graph.

Theorem 2: If is a local Hamiltonian family that lies on an -dimensional lattice, then has a family of low-energy states with low circuit complexity. In particular, if is a local Hamiltonian on a -dimensional lattice acting on qubits for large enough , then for any , there exists a state that can be generated by a circuit of constant depth and such that where is the ground-state energy.

Proof: In what follows, we’ll omit some of the more annoying computational details in the interest of communicating the high-level idea.

Start by partitioning the -dimensional lattice (the one that lives on) into hypercubes of side length . We can “restrict” to a given hypercube (let’s call it ) by throwing away all local terms containing a qubit not in . This gives us a well-defined Hamiltonian on the qubits in . Define to be the -qubit ground state of , and define

where is an -qubit state. Each can be generated by a circuit with at most gates, hence at most depth. Then, can be generated by putting all of these individual circuits in parallel – this doesn’t violate any sort of no-cloning condition because the individual circuits act on disjoint sets of qubits. Therefore, can be generated by a circuit of depth at most . and are both constants, so can be generated by a constant-depth circuit.

We claim that, for the right choice of , is also a low-energy state. Intuitively, this is true because can only be “worse” than a true ground state of on local Hamiltonian terms that do not lie entirely within a single hypercube (i.e. the boundary terms), and by choosing appropriately we can make this a vanishingly small fraction of the local terms of . Let’s work this out explicitly.

Each hypercube has surface area , and there are hypercubes in the lattice. Thus, the total number of qubits on boundaries is at most . The number of size -connected components containing a given point in a -dimensional lattice is a function of and . Both of these are constants. Therefore, the number of size -connected components containing a given vertex, and hence the number of local Hamiltonian terms containing a given qubit, is constant. Thus, the total number of violated local Hamiltonian terms is at most . Taking to be , we get the desired bound. Note that to be fully rigorous, we need to justify that the boundary terms don’t blow up the energy, but this is left as an exercise for the reader.

Linearizing the Hamiltonian

Now that we have shown that Hamiltonians that live on constant-dimensional lattices cannot be used to prove NLTS, we will put the final nail in the coffin by showing that our NLETS Hamiltonian (the Feynman-Kitaev clock Hamiltonian on the circuit ) can be made to lie on a line (a -dimensional lattice). To do so, we will need to understand the details of a bit better.

Proposition 6.1: for the circuit is -local.

Proof: Recall that we defined

Let’s go through the right-hand-side term-by-term. We will use to denote the qubit of the time register and to denote the qubit of the state register.

needs to serially access the qubit pairs for all and ensure that they are all set to . Thus, is -local.

Each term needs to access the states , and and ensure that the state transitions are correct. Thus, is -local.

needs to access the states and ensure that the progression of the time register is correct. Thus, is -local.

Now, we follow an approach of [3] to embed into a line.

Theorem 3: The Feynman-Kitaev clock Hamiltonian can be manipulated into a -local Hamiltonian acting on qutrits on a line.

Proof: Rather than having act on total qubits ( time qubits and state qubits), let’s fuse each and pair into a single qudit of dimension . If we view as acting on the space of particles , we observe that, following Proposition 6.1, each local term needs to check at most the particles corresponding to times , , and . Therefore, is -local and on a line, as desired.

Image from [2]

To see that we can have act on particles of dimension (qutrits) rather than particles of dimension , note that the degree of freedom corresponding to is unused, as the qubit of the state is never nonzero until timestamp . Thus, we can take the vectors

as a basis for each qutrit.

Even though we’ve shown that the clock Hamiltonian for our original circuit cannot be used to prove NLTS (which is still weaker than the original Quantum PCP conjecture) this does not necessarily rule out the use of this approach for other “hard” circuits which might then allow us to prove NLTS. Furthermore, NLETS is independently interesting, as the notion of being low “Hamming distance” away from vectors is exactly what is used in error-correcting codes.

References

[1] Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009.

[2] Chinmay Nirkhe, Umesh Vazirani, and Henry Yuen. Approximate low-weight check codes and circuit lower bounds for noisy ground states. arXiv preprint arXiv:1802.07419, 2018.

[3] Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau, Seth Lloyd, and Oded Regev. Adiabatic quantum computation is equivalent to standard quantum computation. SIAM J. Comput., 2007.

Windows On Theory published first on Windows On Theory

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